ST 351 - Statistical Methods (Combined Second Half Study Set)

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Suppose a market researcher wants to determine what proportion of teens have over 500 Facebook friends. The researcher cannot collect information on every teen, so he obtains a representative sample of all teens throughout the United States using random selection. From the data collected, he estimates the true proportion of teens in the United States who have over 500 Facebook friends. What is this an example of: Estimation Problem or Hypothesis Test Problem?

Estimation Problem

With the increasing popularity of online dating services the truthfulness of information in the personal profiles provided by users is a topic of interest. A study was performed to investigate misrepresentation of personal characteristics. In particular, researchers wondered what proportion of online daters believe they have misrepresented themselves in an online profile. They randomly sampled and contacted 1200 people with profiles in online dating services. Is this scenario an example of an estimation problem or hypothesis test problem?

Estimation Problem

A controversy exists over the distraction caused by digital billboards along highways. Researchers wondered if response time to road signs was greater when digital billboards were present compared to when they weren't present. In a study they conducted to answer their question of interest, 48 people made a 9 km/hr drive in a driving simulator. Drivers were instructed to change lanes according to roadside lane change signs. Some of the lane changes occurred near digital billboards and some did not. What was displayed on the digital billboard changed once during the time that the billboard was visible by the driver. Is this scenario an example of an estimation problem or hypothesis test problem?

Estimation Problem - The key is to identify that there are two groups (i.e. two populations) being compared: drivers who drive past digital billboards and drivers who do not drive past digital billboards

Independent Events

Events are independent if the occurrence of one event does not affect the occurrence of any of the other events • Another way to assess if two events are independent: if the probability of any event remains the same regardless of the occurrence of the other event, the two events are independent.

When comparing two groups, subscripts next to the parameter are needed to distinguish between the two groups. Because the variable of interest is quantitative, the parameter is μ, the mean end-of-year exam scores. Add subscripts to distinguish between the two groups:

Ex. μnew = the mean end-of-year standardized exam score for 4th through 6th graders in this large school district using the new reading program. Ex. μexist = the mean end-of-year standardized exam score for 4th through 6th graders in this large school district using the existing reading program. • Any subscripts can be used. It will be important to define the notation and describe what the parameter and subscript represent in the context of the problem.

Now that the variable of interest is a proportion, the parameter of interest is the population proportion which has notation p. When writing the null and alternative for a categorical variable, keep this parameter in mind.

Example of what the null and alternative hypotheses may look like: H0: p = p0 versus HA: p < p0 or HA: p > p0 or HA: p ≠ p0

True of False? Researchers can make any difference between the observed statistic and the hypothesized value practically significant by taking a large enough sample size, no matter how small that difference may be.

False

True or False? A bootstrap distribution will always be normally distributed.

False

True or False? A confidence interval for a median can be constructed using the formula method.

False

True or False? If the sample size is large enough, the shape of the population will be normal.

False

True or False? Practical significance and statistical significance are the same thing.

False

True or False? Sample means cannot have any variation since sample means are measures of the center and variation refers to the spread of data.

False

True or False? Suppose you had matched or paired data. As long as you have data on each group, you can easily calculate a confidence interval using two-sample t-methods and come out with the same interval.

False

True or False? The smaller the p-value, the weaker the evidence to say the alternative hypothesis is true.

False

True or False? The spread of the bootstrap distribution of sample means is less than the spread of the distribution of sample means.

False

True or False? The null hypothesis, will include inequalities

False

True or False? A simulated experiment should be done under different conditions than the original experiment to create randomness.

False

True or False? The lower bound of a 95% confidence interval is at the 95th percentile, which is the value such that 95 percent of the remaining observations are below it.

False

True or False? The spread of the sample means can be greater than the spread of the data in the population.

False

True or False? Practical significance is the same as statistical significance.

False • Sometimes statisticians find statistical significance from data collected but this evidence is not actually useful in a real world context. Therefore, practical significance is not the same as statistical significance.

True or False? T-methods cannot be used if the sample size is less than 30.

False • The t-methods rely on the sampling distribution of the sample mean being normal. We know that the sampling distribution of the sample mean will have a normal shape when one of two things occur, the sample size is large OR the population data has a normal shape.

True or False? We adjust the bootstrap sample values to center the bootstrap distribution of the sample means at the hypothesized value in order to determine the p-value.

False • We adjust the original sample values before bootstrapping takes place. That is, we bootstrap from the adjusted sample not from the original sample.

True or False? In hypothesis testing we collect data to determine if we have evidence against the alternative hypothesis

False - A conclusion from a hypothesis test is a statement of how much evidence we have to support the alternative hypothesis based on what we observed in the sample. Therefore, this statement is false

True or False: 95 is an example of how a confidence level is written.

False - Confidence levels are generally written as percentages, such as 95%. The only other method of reporting a confidence level is as a proportion, such as 0.95. It is never reported as an integer value without a % symbol next to it.

True or False? All bootstrap statistics will be the same since they are taken from the same sample.

False - Since bootstrap samples are obtained by sampling from the original sample with replacement each bootstrap sample is different and therefore it is possible that bootstrap sample statistics will be different. Although there may be repetition in the actual values of the bootstrap statistics there is no guarantee that they will all be the same.

True or False? The population data will be approximately normal only if the sample size is large enough

False - population is fixed. The data will have a particular distribution before a sample is taken and it will not change once a sample is taken

True or False? The spread of the distribution of sample means increases as the sample size increases.

False - the spread of the distribution of sample means decreases as the sample size increases

True or False? The shape of the distribution of sample means will never be normally distributed if samples of size 10 are taken.

False - when the population data are normally distributed, the distribution of sample means will be normal regardless of the sample size

To find the p-value using t-methods:

First: Convert the observed sample mean to a t-statistic Second: Determine the degrees of freedom for the t-statistic Finally: Using RStudio, determine the p-value

Notation for the Significance Level:

Greek letter alpha: α

Null Hypothesis notation:

With a single quantitative or categorical variable, the null hypothesis is written in notation as follows:

H0: parameter = hypothesized value

H0: μ = 800

Notation for comparing a quantitative variable between two groups, when the null hypothesis is that the means of the two groups are the same:

H0: μgroup1 = μgroup2 OR H0: μgroup1 − μgroup2 = 0

With a single quantitative or categorical variable, the alternative hypothesis is written in notation in one of the following ways:

HA: parameter > hypothesized value HA: parameter < hypothesized value HA: parameter ≠ hypothesized value

HA: μ < 100 miles

When comparing the mean between two groups, the alternative hypothesis will be written in one of the following ways:

HA: μgroup1 − μgroup2 < 0 HA: μgroup1 − μgroup2 > 0 HA: μgroup1 − μgroup2 ≠ 0

What is the sample space of a coin flip?

HH, TT, HT, TH

Three confidence intervals for the mean amount spent per week on groceries for a family of four are given below. Rank the confidence intervals from highest level of confidence to lowest level of confidence. confidence interval #1: ($120, $140) confidence interval #2: ($115, $145) confidence interval #3: ($125, $135) Highest level of confidence: Second highest level of confidence: Lowest level of confidence:

Highest level of confidence: confidence interval #2 Second highest level of confidence: confidence interval #1 Lowest level of confidence: confidence interval #3

Margin of Error

How far away the sample statistic (such as the sample mean) can be from the population parameter with a certain level of confidence Ex. +/- 3%

Researchers wondered if parents or their teen children use social media more, on average. A random sample of 1000 teens and their parents was taken. Each was asked how many times a day they check social media sites. Is this scenario an example of an estimation problem or hypothesis test problem?

Hypothesis Test

The Equally Likely Formula to calculate a probability:

If (and only if) the outcomes in the sample space are equally likely to occur, the following formula can be used to calculate the probability of an event of interest. Let A = an event of interest. P(A) = # of outcomes in the event of interest / number of outcomes in the sample space

Statistically Significant

If a decision is made to reject the claim in the null hypothesis

Statistical significance

If the decision is made to reject the claim made in the null hypothesis for the claim made in the alternative hypothesis • Statistical significance is not the same as practical significance

Why is it important to sample with replacement?

If the sample is representative of the population, each randomly selected person's value will represent many other people in the population who have that same value.

Probability Rule #3: The Complement Rule

If two events are complements, the probability of the complement of an event = 1 - the probability of the event. • Notation: A^c = complement of A and means "NOT" A. In notation, P(A^c) = 1 - P(A)

Probability Rule #2: The Addition Rule

If two events are disjoint (or "mutually exclusive), then the probability of one or the other occurring is the sum of their separate probabilities. • In other words, if event A and event B are disjoint, P(A or B) = P(A) + P(B), where "or" means either A occurs or B occurs.

Probability Rule #1: The Multiplication Rule

If two events are independent, the probability of both occurring in the same "situation" can be found by multiplying the probabilities of each event together. • In other words, if event A and event B are independent, P(A and B) = P(A)*P(B), where "and" means both occur in the same "situation", and * means to multiply.

Why is it important that the distribution of sample means be approximately normal?

In order to use the t-methods, the distribution of sample means MUST be approximately normal

Central Limit Theorem

In some situations, when independent random variables are added, their properly normalized sum tends toward a normal distribution, with standard deviation decreasing, even if the original variables themselves are not normally distributed • If the population distribution for the variable X has a mean of μ and a standard deviation of σ, then the distribution of sample means becomes closer and closer to a normal distribution with mean, μx̅ , equal to the population mean and standard deviation equal to σx̅ = σn/√σx as the sample size increases

The Central Limit Theorem

The theory that, as sample size increases, the distribution of sample means of size n, randomly selected, approaches a normal distribution. • If a sufficiently large random sample of size n is drawn from ANY population with mean µ and standard deviation σ, the distribution of the sample means is approximately normal with mean μx̅ =μ and standard deviation σx̅ = σ/√n no matter the shape of the data in the population.

A basketball player shoots two free throws in a row. Let A = she makes the first free throw. Let B = she misses the second free throw. Are the two events disjoint?

The two events are not disjoint since both of these events can happen in the same "experiment" (the two free throw attempts). That is, she can make the first attempt (event A) and miss the second attempt (event B).

The researcher has to decide ahead of time the probability he/she wants to have of making an error if the decision is to reject the null hypothesis. That is, before the study begins, the researcher decides the probability of making a Type __ Error.

Type I Error

A decision is made to fail to reject the null hypothesis. What error could be made? Type I Error or Type II Error

Type II Error

Suppose that in a jury trial, the jury gives the judge the verdict "not guilty". What type of error could the jury have made? Type I Error or Type II Error

Type II Error • The decision "not guilty" is failing to reject the null hypothesis. Therefore, the jury could be making a Type II Error.

Standard Error

Used to refer to the standard deviation of the sampling distribution.

Percentile Method

Uses the fact that a confidence interval contains the middle part of the sampling distribution Ex. The values in a 95% confidence interval are the middle 95% of the sampling distribution

The owners of a small gardening company are interested in the average time it takes a rose in Portland to reach full bloom so that they can determine how early they should begin taking orders for their rose selling business. They petition 400 randomly selected residents of Portland to watch the roses in their garden and report how long (in weeks) it takes for each rose in their rosebushes to reach full bloom. They believe that the average time it takes a rose in Portland to reach full bloom is less than 8 weeks. What is the variable of interest? What is the population of interest?

What is the variable of interest? The time it takes for roses to reach full bloom in weeks What is the population of interest? Roses in Portland

The General Multiplication Rule:

When events are dependent, we can still multiply probabilities together. But, the probabilities multiplied together will be their conditioned probabilities.

Proper syntax for writing confidence interval:

When reporting a confidence interval, we report it as follows: (lower bound, upper bound) • Both bounds appear inside parentheses separated by a comma • The lower bound is always reported first. • Always include units!

The width of a confidence interval is the difference between the upper and lower bounds:

Width of confidence interval = upper bound - lower bound

a) hypothesis test problem • Since there is a particular value for the average the store owners can compare to ($800 from last year), a hypothesis test can be performed

State if each of the following is an example of a point estimate, interval estimate, parameter, or none of these. a) means from 90 to 100 b) the list of all responses to a yes/no question in a poll

a) interval estimate b) none of these

a) one-sided

State if each of the following is an example of a point estimate, interval estimate, parameter, or none of these. a) a sample proportion b) a population mean c) the median of a data set from a sample

a) point estimate b) parameter c) point estimate

Match the scenario with the type (sampling with/without replacement) of sampling: a) Each card is put back into the deck before the next card is drawn b) Each card is not put back into the deck before the next card is drawn

a) sampling with replacement b) sampling without replacement

The ______ the p-value, the stronger we would feel about saying the null hypothesis is NOT true and the alternative hypothesis IS true. a) smaller b) greater

a) smaller • Remember, when the p-value is LOW reject the H-0

In order to use the t-methods, the distribution of sample means must be __________. a) symmetric and bell-shaped b) unimodal and left skewed c) unimodal and right skewed

a) symmetric and bell-shaped • In order to use the t-methods for hypothesis testing, we must have a normal sampling distribution. It is one of the conditions of the test.

Before collecting any data, we start by believing _____ is true a) the null hypothesis b) the alternative hypothesis c) neither the null nor the alternative hypothesis

a) the null hypothesis

For hypothesis tests on a single variable, the hypothesized value is ______ for both the null and alternative hypotheses a) the same b) different

a) the same

If an outcome can never occur, P(outcome) equals?

Choose the appropriate adjective for each of the following p-values: a. Strong or convincing b. Some c. Weak d. Not sufficient 1. p-value = 0.0001 2. p-value = 0.1295 3. p-value = 0.99

1. a. Strong or convincing 2. d. Not sufficient 3. d. Not sufficient

Fill in the blanks to complete how a confidence level is determined. A 90% confidence interval implies that ___% of ___ unique random samples would have confidence intervals that capture the _____ (assuming all such confidence intervals were constructed in the same manner).

90%, all, parameter

Estimation

A best guess as to what the population problem is

Null Hypothesis

A claim that we believe as true to start with • This is often consistent with an idea of no change or no difference

Critical Value t-distribution formula:

CL+[(1−CL)/2] where CL is the confidence level reported as a proportion Ex. If you wish to obtain a critical value for a 95% confidence interval, the percentile is 0.95 + [(1−0.95)/2] = 0.975

_____ letters near the ______ of the alphabet are generally used to notate an event of interest

Capital, beginning Ex. A = getting one head and one tail on two tosses of a coin

Complement notation:

If A is an event, the complement of A is notated as A^c

21. According to the central limit theorem, as sample size increases, the standard deviation of the sampling distribution of sample means a. does not change. b. decreases. c. increases. d. Impossible to know.

b. decreases.

17. For a normal population, what can we say about the sampling distribution of the sample mean? a. The sampling distribution will only be normal as long as the sample size is large. Otherwise the sampling distribution will be skewed. b. For any sample size, the sampling distribution will be normal with mean equal to the sample mean (𝜇x̅ = x̅) and standard deviation 𝜎x̅ = 𝜎. c. For any sample size, the sampling distribution will be normal with mean equal to the population mean (𝜇x̅ = 𝜇) and standard deviation 𝜎x̅ = 𝜎√𝑛. d. For any sample size, the sampling distribution will be normal with mean equal to the population standard deviation (𝜇x̅ = 𝜎) and standard deviation 𝜎x̅ = 𝜇. e. All of the above statements are false.

c. For any sample size, the sampling distribution will be normal with mean equal to the population mean (𝜇x̅ = 𝜇) and standard deviation 𝜎x̅ = 𝜎√𝑛.

James has a deck of 52 playing cards. A fair deck of cards has an equal number of red and black cards (26 of each). James's friend, Grace, wonders if the deck is unfair, meaning there are more of one color than the other. To determine if the deck is not fair, cards are drawn one at a time with replacement. What is the parameter that will be in the hypotheses and what does it represent? a) pred and pblack, the proportion of cards in the deck that are red and black, respectively. b) x̅, the average number of cards in the deck that are red c) pˆ, the proportion of cards in the deck that are red d) μ, the average number of cards in the deck that are red e) p, the proportion of cards in the deck that are red

e) p, the proportion of cards in the deck that are red • p is the notation for the population proportion of red cards in the deck

The Oregon Department of Transportation is interested in the average number of miles driven by working adults aged 35 - 46 in a one week period. ODOT believes that working adults in this age range drive more than 100 miles in a one week period. ODOT sends a survey to 1000 such adults and surveys are returned by 274 adults. The survey asks respondents for an estimate of how many hours they believe they drive each week. ODOT determines that the average number of miles driven each week by the respondents of the survey is 74.9 miles. What sample means are "as or more unusual" if the null hypothesis is true? a) ≥ 74.9 miles b) between 74.9 and 100 miles, inclusive c) ≤ 100 miles d) ≥ 100 miles e) ≤ 74.9 miles

e) ≤ 74.9 miles

β (probability of making a Type II error) depends on the _____ _____, which is determined by the researcher.

effect size

A researcher understands that going into a hypothesis test there is always a chance that an _____ could be made with the decision they make.

error

Once we have established that a scenario involves inference, the next step is to identify if the inference is an ______ problem or if it involves a ______ test.

estimation, hypothesis

After hypotheses are determined, the ______ is presented

evidence

If you are constructing a confidence interval using the ______ method, you can use the standard deviation of the adjusted bootstrap distribution as the value of the standard error. There is no need to create another bootstrap distribution in this situation.

formula

The ______ _____ is determined from the question of interest and often is consistent an idea of no change or no difference

hypothesized value

A deck of playing cards has 52 cards, divided into four suits (diamonds, spades, clubs, and hearts). There are an equal number of cards in each suit in this deck. Two cards are drawn from the deck. Let event A = the first card drawn is a club. Let event B = the second card drawn is a club. If we sampled with replacement, are events A and B dependent or independent??

independent

Populations are said to ______ if the sampled cases are from different, unrelated populations.

independent • In other words, we obtain a random sample from Population 1 and another random sample from Population 2.

Suppose we want to determine the average high temperature for all cities in the United States yesterday. We take a random sample of 25 cities and determine the average high temperature of those 25 cities yesterday. Suppose the average high from the sample of 25 cities was x̅ = 72.3ºF. Also suppose the margin of error for a 95% confidence interval was reported as 6.5ºF. Calculate the lower and upper bounds of the 95% confidence interval for the population mean high temperature of all U.S. cities yesterday. Report each to ONE decimal place.

lower bound = 72.3 - 6.5 = 65.8ºF upper bound = 72.3 + 6.5 = 78.8ºF

The smaller the P-value, the _____ evidence there is to say the alternative hypothesis is true.

Coach S.P. Dee knows his team's runners of the 100-meter event will need to run faster than 12 seconds, on average, for their team to have a chance to repeat as conference champions at the end of the year. He records the times of the 100-meter event for his team's runners at three different track meets towards the end of the year. How many populations are there? a) One - all runners b) One - all sprinters of the 100-meter dash on Coach Dee's track team c) One - all sprinters of the 100-meter dash on Coach Dee's track team that run the 100-meter dash faster than 12 seconds. d) Two - all sprinters of the 100-meter dash on Coach Dee's track team that run the 100-meter dash faster than 12 seconds and of the 100-meter dash on Coach Dee's track team that run the 100-meter dash 12 seconds or slower. e) More than two

b) One - all sprinters of the 100-meter dash on Coach Dee's track team • Coach Dee is only interested in making conclusions to sprinters of the 100-meter dash on his team

H0: μ = 10 HA: μ ≠ 10 90% confidence interval for μ: (15, 20) Using the confidence interval, what decision should be made? a) Fail to reject the null hypothesis b) Reject the null hypothesis

b) Reject the null hypothesis • Since the hypothesized value is not between the bounds, we can be confident it is not one of the values the parameter can be. Therefore, we have evidence to reject the null hypothesis.

In general, the null hypothesis is a statement of no _____or no _____.

no difference or no effect

Suppose α = 0.10, β = 0.13, and p-value = 0.03 What decision was made? a) Accept the null hypothesis b) Reject the null hypothesis c) Fail to reject the null hypothesis

b) Reject the null hypothesis • Since the p-value < , the decision is to reject H0.

Anytime we have information on an entire population, ___ _____ is being made because we already know what is happening in the population

no inference

Any reference to a t-distribution requires that the sampling distribution of the statistic be _____. If the populations are _____, the resulting distribution of differences in sample means will also be _____ no matter the sample size.

normal

If the population data are symmetric, the distribution of sample means will be _____ regardless of the sample size.

normal

The distribution of sample means will be ______, or at least approximately ______ if at least one of the following two situations occur: 1. The sample size must be sufficiently large, no matter the shape of the population. Typically sample sizes that are 30 or more are considered large enough. 2. Although the Central Limit Theorem does not specifically mention this situation, if the data in the population are normal, the sample means will be normal regardless of the sample size.

normal

With one-sample z-methods, there is also a condition that the distribution of sample proportions must be _____.

normal • This will occur if the sample size is large enough, but whether the sample is considered large enough depends on what we believe the population proportion to be.

If a study was being conducted to test the effectiveness of a new drug compared to the old drug, the _____ hypothesis would be there is no difference in the effectiveness between the new and old drug.

null

The hypothesized value is the same as in the _____ hypothesis and is determined from the question of interest

null

When the goal is to find a p-value, the bootstrap distribution must be centered at the _____ hypothesized value.

null

The definition of the p-value indicates that there is still a probability of observing what we did (or something more unusual) if the _____ hypothesis is true. A probability indicates it could happen.

null Ex. We may wonder if a deck of cards is "fair". (A "fair" deck would have half red cards and half black cards.) The hypotheses are: • H0: the deck is fair • HA: the deck is not fair.

It is typical that researchers hope that the data lead to a conclusion of rejecting the claim made in the _____ hypothesis in favor of the claim made in the _____ hypothesis.

null, alternative

Parameter notation when you have a single categorical variable of interest and ONE population:

p = population proportion σp̂ = standard deviation for the distribution of sample proportions

In a statistical hypothesis test, the sample will hopefully provide evidence that the alternative hypothesis is true. We can quantify this concept of evidence by finding a _____

p-value

Estimation of P-Value formula:

p-value = count of number of statistics as or more unusual if Ho is true / total number of randomizations generated

Suppose fifty elephants were weighed and a sample mean of 244 pounds was found with a sample standard deviation of 11 pounds. If researchers were interested in constructing a confidence interval for true mean weight of all elephants, could t-methods be used? a) No - without knowing the population mean, it is impossible to construct a confidence interval with any methods. b) Yes - the sample size is considered large enough for the distribution of sample means to be normal. c) No - the sample standard deviation is not more than 30, not considered large enough, for the distribution of sample means to be normal. d) Yes - the sample mean is considered large enough for the distribution of sample means to be normal.

b) Yes - the sample size is considered large enough for the distribution of sample means to be normal. • Since the sample size for this problem is 50, we would consider this large enough to assume the shape of the distribution of sample means is approximately normal.

Inference means to use ______ to make a conclusion about the population from which the sample came. a) the mean of the sample b) data from a sample c) specific members of a sample

b) data from a sample

The sample statistic ______ appear in the hypotheses. This is because the hypotheses are determined before any sample data are collected a) does b) does not

b) does not

To obtain a higher level of confidence that a confidence interval captures the true population mean, the interval a) will always contain the same number of values b) must contain more values c) must contain fewer values

b) must contain more values

b) one: the cards in the deck • The question is about the deck of cards so that is our population

If p^ is a sample proportion from a random sample of size n and if n is sufficiently large, then a confidence interval for the population proportion has the formula:

p^ ± (z∗) (√p^(1−p^)/n) • You must use po when calculating the standard error and when you are constructing a confidence interval, you must use p^ in the standard error.

Data is considered _____ when either one individual has two measurements on them and they may act as his or her own control or two individuals have been matched on numerous characteristics.

paired

Recognize that when the problem requires us to compare the means between two groups in an alternative hypothesis, the _____ is the difference in population means.

parameter

A ______ is a number that describes some aspect of a population. A ______ is a number that is computed from data in a sample.

parameter, statistic

When two-sample t-methods are used, researchers are asking the question if there is a difference in _____ means. This implies that our parameter of interest is the difference in _____ means, μ1 − μ2.

population

The parameter will be the _____ _____, when the variable is quantitative and we believe the population data is reasonably symmetric

population mean (μ)

The parameter will be the _____ _____, for a categorical variable with two categories

population parameter (p)

If the difference between the observed statistic and hypothesized values is less than this "effect size", the researcher may say that the data are not "_____ _____" (i.e. there is practically no difference between the observed statistic and the hypothesized value).

practically significant

It is sometimes hard to assess the shape of sample data when the sample size is small. A normal ______ plot can be used to assess whether data are normal for any sample size.

probability

Probabilities are ______, so we will write the percentage as a ______

proportions, proportion

Statistics/Sample notation when you have a single categorical variable of interest and ONE population:

p̂ = sample proportion SEp̂ = standard error of the sample proportion

Flipping a coin, rolling a die, drawing a card from a deck, and even taking a random sample, are examples of _____ _____ in the _____ sense

random experiments, probability

A Type II Error can only be made when a decision is made to _____ the null hypothesis.

reject

If the hypothesized value does NOT fall between the bounds of a confidence interval, the decision is to ______ the claim made in the null hypothesis in a two-sided test. Reject, fail to reject, or accept?

reject • Since a confidence interval is a list of all possible values of the population parameter could be, then if our hypothesized value is outside the interval we are essentially saying it is not a possible value for the parameter. We can reject the null hypothesis.

If the sample is _______ of the population, we can think of each case in the sample as representing many cases in the population with that same value

representative

To use one-sample z-methods, we must have a _____ sample with _____ observations.

representative, independent

The approach of making a decision based on a pre-determined significance level is called the ______ ______ approach to making a decision.

significance level

If the data in the population are heavily skewed and/or there are outliers, much larger sample sizes are needed in order for the sample means to be approximately normal. The sample size needed in such a situation could range from as low as 50 up to a thousand. The more heavily _____ the population data, the _____ the sample size needed.

skewed, larger

The only time the distribution of sample means will not be approximately normal is if the sample size is _____ and the data in the population is _____.

small, skewed

A t-statistic is the number of _____ _____ a sample mean is from the null hypothesized value for the population mean, µ0

standard errors

statistic - Since the value 9.4 was calculated using only a sample of 30 miners, therefore it is considered a statistic

Researchers can make any difference between the hypothesized value and the observed statistic "_____ significant" just by taking a large sample size. However, that difference may be very small - in a sense it's not "_____ significant."

statistically, practically

When doing inference problems with samples, we will use a ______ instead of a standard normal distribution.

t-distribution

When it comes time to find a p-value from a two-sample t-test, we will need to reference a ______.

t-distribution

When the population standard deviation is not known, we will use a ______ for inference problems instead of a standard normal distribution.

t-distribution

If you feel comfortable saying that the distribution of sample means is approximately normal, then you should feel comfortable using ______ to find your p-value.

t-methods

Two Sample t-Statistic formula:

tdf = (observed statistic − hypothesized value) / standard error • Since we start with the belief that the null hypothesis is true when hypothesis testing, we have μ1−μ2 = 0.

To test the hypothesis H0 : µ = µ0 against any alternative hypothesis, we can compute the one-sample t-statistic by this formula:

tn−1 = x̅−μ0/s√n

The null hypothesis is not what the researcher is trying to prove, but it is the claim we will assume to be _____ before collecting any data a) true b) not true

true

he technology committee at a university has stated that the average time spent by students per lab visit has increased. Students are spending more time in the computer labs which means there needs to be more funding (through increased lab fees). To substantiate this claim, the committee randomly samples 25 students who have visited the lab and records the amount of time spent using the computer. The mean time spent on the computers was 63.6 minutes with a standard deviation of 8.07 minutes. Using the information, report the following values: x̅ = __ minutes s = __ minutes n = __

x̅ = 63.6 minutes s = 8.07 minutes n = 25

Statistics/Sample notation when you have a single quantitative variable of interest and TWO populations:

x̅1-x̅2 = difference in sample means s1-s2 = standard deviation of each sample SEx̅1-x̅2 = standard error of the difference in means

Formula for "constructing" a confidence interval for a population mean:

x̅± some amount • The "some amount" is called the margin of error.

Formula for constructing a confidence interval for μ:

x̅±(t∗n−1)(s√n) t∗n−1 = critical values

Confidence Interval for the Population Mean formula:

x̅±(z−score)(SEx̅) Therefore, there are three pieces of information we need to construct the confidence interval: x̅, the sample mean, the z-score, and SEx̅, the standard error of the distribution of sample means

One Sample Z-Statistic Formula

z = (p^−po) / √(po(1−po)n • p^ = observed • po = predicted • n = sampel size

The critical values used for a confidence interval for a population proportion are ______ values.

z-critical • Since z-critical values do not depend on any degrees of freedom, the values you will need to consider will only be based on the level of confidence you wish to have

Once you discover you are dealing with categorical data, the next step would be to see if the conditions necessary to use one-sample ______ have been met.

z-methods

A game show host selects participants from an audience of 60 people to participate in the game. For each round a different participant is randomly selected from the audience. Once an audience member from one row is selected no one from that row can be selected. Audience members are seated in rows of 10. What is the probability that an audience member in seat 25 is selected given an audience member in seat 23 was already selected.

zero

Significance Level formula:

α (as a percentage) = 100 - confidence level

Notation for the probability of making a Type II Error:

β (the Greek letter beta)

μ < 12 seconds

Parameter notation when you have a single quantitative variable of interest and ONE population:

μ = population mean σ = population standard deviation μx̅ = mean of the distribution of sample means SEx̅ = s/√n = standard error of the mean

Parameter notation when you have a single quantitative variable of interest and TWO populations:

μ1-μ2 = difference in population means σx̅1-x̅2 = standard deviation of the difference in means

Mean of Sample Means notation:

μx̅

Standard Deviation of Sample Means notation:

σx̅

Comments about the distribution of sample means center:

• A distribution of sample means is always centered at the population mean. In fact, μx̅ = μ

b) p = the proportion of candies that are orange in a bag of M&M's milk chocolate candies. • The variable of interest is whether or not a piece of candy is orange, which is categorical with two categories (yes it is or not it is not). Therefore, the parameter is the population proportion in one of the two categories.

A _____ is a value of a statistic that best estimates a population parameter. a) mean b) point estimate c) population parameter

b) point estimate

Based on correctly simulating the Monty Hall problem, which strategy would you recommend to the contestant? a) stay b) switch c) it doesn't matter

b) switch - While it may not seem intuitive, a contestant will win the car 2/3 of the time they choose to switch but only 1/3 of the time they choose to stay. Another method of determining these probabilities will be shown later in this learning path.

In a statistical hypothesis test, the evidence is a) the data from the entire population b) the data from the sample, other observational study, or experiment c) information the researcher collects to support his/her belief

b) the data from the sample, other observational study, or experiment

When comparing a quantitative variable between two groups, the null hypothesis is that the means of the two groups are a) different b) the same

b) the same

b) two-sided • Since the alternative hypothesis contains a ≠, we will perform a two-sided test

Here are the summary statistics for the original sample of 5 students: min 6 Q1 11 median17 Q3 22 max 29 mean 17 sd 9.027735 n 5 missing 0 1. What is the correct notation and value for the mean of the 5 values in the sample? a) μx̅ = 17 months b) x̅ = 17 months c) μ = 17 months

b) x̅ = 17 months

Which of the following is the correct symbol for a significance level? a) p (for p-value) b) α c) x̅ d) a e) μ

b) α

The U.S. Census Bureau reports that the average number of employees in a small business is 16.1. A random sample 45 small businesses are contacted and were asked how many employees were on their payroll. A sample mean of 17.6 employees was found. 15. What is the approximate probability of observing a sample mean of 17.6 or greater? (Hint: can you use the 68-95-99.7 rule here. Double hint: Yes you can!) a. 0.0015 b. 0.025 c. 0.16 d. 0.95 e. 0.975

b. 0.025

16. Penny has just performed a one-sample t-test and found a one-sided p-value of 1.05. What advice do you have for Penny in regards to this p-value she has found? a. Since the p-value is greater than 1, Penny actually performed a two-sided test. She should divide this 1.05 by 2 to obtain the correct p-value. b. Penny made a mistake somewhere because it is not possible to get a p-value greater than 1. c. A one-sample t-test cannot produce a p-value greater than 1, but a hypothesis test using bootstrapping methods could. She must have used bootstrapping methods. d. Penny should take this p-value and subtract it from 1, resulting in the correct p-value of 0.05

b. Penny made a mistake somewhere because it is not possible to get a p-value greater than 1.

22. A study was conducted which looked at parental empathy for sensitivity cues and baby temperament (higher scores mean more empathy). It was of interest to compare the average empathy score between mothers and fathers. A random sample of mothers and a random sample of fathers were gathered. The table below provides summary statistics for the study: Mothers: Sample size 42 Sample mean empathy score 69.44 Sample standard deviation 11.69 Fathers: Sample size 45 Sample mean empathy score 59 Sample standard deviation 11.60 Which of the following tests should researchers use to test if there is a difference in the mean empathy scores between mothers and fathers? a. One sample t-test b. Two-sample t-test c. Two-sample z-test for proportions d. One-sample inference using bootstrapping methods.

b. Two-sample t-test

19. When is the only time it is even possible to make a Type II Error? a. When we fail to reject the alternative hypothesis b. When we fail to reject the null hypothesis c. When we reject the null hypothesis d. When we accept the null hypothesis

b. When we fail to reject the null hypothesis

A biologist is interested in studying the effects of applying insecticide to a fruit farm on the local bat population. She collects a random sample of 30 bats from each of the 5 major fruit farms in this area (150 bats total) and finds a mean weight from this sample to be 150.4 grams. Previous research suggests that bats in this area should have a mean weight an average of 157 grams. 12. If the null hypothesis is true, the probability of observing a sample mean 150.4 or less was found to be 0.006. This probability is known as? a. the standard error b. the p-value c. the compliment d. the significance level e. the probability of a Type II error

b. the p-value

Conditions of the Two-Sample t-Test:

• A quantitative variable of interest is measured on all cases. • Two samples were randomly selected from two independent populations • Data in each sample is representative of data in respective populations • The distribution of differences in sample means x̅1 − x̅2 is normally distributed. This will occur if: The data in each population is normally distributed and/or each sample size is large. The Central Limit Theorem then supports assuming the distribution of differences in sample means will be normal. Recall that the sample size needed depends on how skewed the population data are -- the more skewed the population data, the larger the sample size needed for the distribution of differences in sample means to be normal.

There are several key things to look for to tell if a scenario involves inference or not:

• Does the scenario mention a sample was taken? If so, there is a good chance the scenario involves making an inference. • Does the scenario mention a plan to estimate a parameter of interest? • Does the scenario make any sort of claim about a mean, median, or proportion? In such a situation, we may be wanting to make a decision about that claim in the population, which involves making an inference. • Does the scenario involve comparing two or more groups where the researchers want to make a comparison between these groups? If so, the scenario most likely involves making an inference to the populations.

From the 68-95-99.7 rule, approximately 95% of the data would fall within two standard deviations of the mean of its distribution IF the data are normally distributed:

• For example, if we construct a bootstrap distribution of sample means, the data are the bootstrap sample means • The bootstrap distribution of sample means is centered at the sample mean. • Therefore, if the bootstrap distribution of sample means is normal, approximately 95% of the bootstrap sample means will be within 2 standard deviations of the sample mean • These two values that are two standard deviations from the sample mean are the lower and upper bounds of a 95% confidence interval constructed using the formula method (x̅±(z-score)(SEx̅)), where SEx̅ is the standard deviation of the bootstrap distribution • However, these two values that are the lower and upper bounds are also values of certain percentiles of the bootstrap distribution

How to decide which significance level to use:

• For studies where it is important to feel nearly certain that the alternative hypothesis is true, such as in medical studies, the significance level may be set at 0.01 • For studies where it is not as crucial to be completely sure that the alternative hypothesis is true, the significance level might be set at 0.10 • The most typical significance level is in between these two extremes and is 0.05. Most areas of science use this significance level

Things to remember for types of research study problems:

• For those longer research problems: - Determine what kind of data you are working with (categorical or quantitative) and how many populations you have (1 or 2) • For quantitative data - if information is given, determine is the sample is symmetric or skewed: - If skewed or extreme outliers - use the median as measure of center - If symmetric - use mean as measure of center

Notation: Hypothesis Testing One-Sample Tests -> One Population:

• H0: the null hypothesis: H0: μ = μ0 or H0: p = p0 • HA: the alternative hypothesis can have one of the following forms: • HA: μ = μ0 -> two-sided p-value or HA: p ≠ p0 -> two-sided p-value • HA: μ < μ0 -> one-sided p-value or HA: p > p0 -> one-sided p-value • HA: μ > μ0 -> one-sided p-value or HA: p > p0 -> one-sided p-value • μ0 and p0 are null hypothesized value for either μ or p. This should be a number.

Notation: Hypothesis Testing Two-Sample Test for the difference in population means -> Two Populations:

• H0: the null hypothesis: H0: μ1 = μ2 or H0: μ1 - μ2 = 0 • HA: the alternative hypothesis can have one of the following forms: • HA: μ1 - μ2 ≠ 0 -> two-sided p-value • HA: μ1 - μ2 < 0 -> one-sided p-value • HA: μ1 - μ2 > 0 -> one-sided p-value

P-Value Significance Level Summary:

• If a p-value ≤ α, the decision is to say the alternative hypothesis is true. This is equivalent to saying we reject the null hypothesis. • If a p-value > α, the decision is to NOT say the alternative hypothesis is true and we, therefore, fail to reject the null hypothesis. This is not the same thing as saying the null hypothesis is true.

What method to use when constructing a confidence interval:

• If the bootstrap distribution is heavily skewed, do not use either method - If your variable is quantitative, you can construct a confidence interval for the median - You can only construct a confidence interval for the median using the percentile method • If the bootstrap distribution is slightly skewed, use the percentile method • If the bootstrap distribution is symmetric but does not have an approximate bell shape, use the percentile method • If the bootstrap distribution is symmetric with a bell shape, either method can be used

Summary of how the confidence interval can be used to make a decision about the null hypothesis:

• If the hypothesized value falls between the bounds (inclusive), the decision is not to reject the claim made in the null hypothesis as the hypothesized value is one of the possible values the parameter could be with a certain level of confidence. • If the hypothesized value does NOT fall between the bounds (inclusive), the decision is to reject the claim made in the null hypothesis as the hypothesized value is not one of the possible values the parameter could be with a certain level of confidence. • These decisions are based on a significance level that is the complement of the confidence level. - That is, α (as a percentage) = 100 - confidence level

What exactly is bootstrapping?

• It is a procedure that allows us to estimate the true sampling distribution. • We take the sample with replacement from our original sample and calculate a statistic, such as mean or proportion. We repeat this thousands of times which results in thousands of bootstrapped statistics. • A sampling distribution is a distribution of a statistic calculated from every possible unique random sample, of the same size, from some population. • A sampling distribution is needed so that we can understand how statistics vary from one sample to the next. The measure of spread of these statistics is described by the standard deviation of the sampling distribution (σx̅ or σp̂ or σx̅1-x̅2).

Summarizing the Properties of a Bootstrap Distribution of Sample Means: Spread:

• Just like with the sampling distribution, - the spread of the bootstrap distribution will be less than the spread of the sample data. - the spread of a bootstrap distribution decreases when a larger sample size is taken. Please note that the sample size is not the same as the number of bootstrap samples you generate.

Summarizing the Properties of a Bootstrap Distribution of Sample Means: Shape:

• Like the distribution of sample means, a bootstrap distribution of sample means: - will be normally distributed if the population data are normally distributed - will be approximately normally distributed if the population data are skewed and the sample size is sufficiently large. The more skewness in the population data, the larger the sample size needs to be for the bootstrap distribution to be approximately normal

Comments about the distribution of sample means shape:

• Normal: - If the population data are normally distributed regardless of the sample size • Approximately Normal: - If the population data are skewed AND the sample size is sufficiently large, where this may depend on how skewed the population data are - For slightly skewed population data, a sample size of at least 30 is typically considered large enough - For extremely skewed population data the sample size may need to be in the hundreds - Note that there is no set minimum sample size that works for every situation

Medical researchers hope to find a treatment that is "better." What is the null and alternative hypothesis?

• Null hypothesis is that there is no difference between the new treatment and old treatment. • Alternative hypothesis is that there is a difference, which is what the researchers hope to find.

Sample Distribution spread notation:

• Quantitative data: population standard deviation notation: s • Categorical data: there is no spread.

Population Distribution spread notation:

• Quantitative data: population standard deviation notation: σ • Categorical data: there is no spread.

A bootstrap distribution of sample means will be skewed when:

• Sample sizes are small and the population data are lightly to heavily skewed • Sample sizes are large and the population data are heavily skewed

Significance Levels:

• Significance levels are determined at the start of a study, along with the hypotheses. • Significance levels are set because researchers want the probability of wrongfully rejecting the null to be more than α. - That is why in order to reject H0 our p-value must be less than α.

TRUE Sampling Distribution formula & notation:

• Standard deviation of the distribution of sample means: σx̅ = σ/√n • Standard deviation of the distribution of differences in sample means: σx̅1-x̅2 = √(σ²1/n1)+(σ²2/n2) • Standard deviation of the distribution of differences in sample proportions: σp̂ = √p(1-p)/n

Standard Deviation/Standard Error formula & notation used in practice:

• Standard error of the mean. Use to estimate σx̅: SEx̅ = s/√n • Standard error of the difference in means. Use to estimate σx̅1-x̅2: σx̅1-x̅2 = √(s²1/n1)+(s²2/n2) • Standard error of the sample mean. Use to estimate σp̂ in confidence interval calculations: SEp̂ = √p̂(1-p̂ )/n

Summarizing the Properties of a Bootstrap Distribution of Sample Means: Center:

• The bootstrap distribution will be centered at the original sample mean. - The properties given for the distribution of sample means included that the mean of the distribution of sample means equals the population mean, μ. Hopefully, the sample mean is close to the population but we are sure it will not equal the population mean perfectly. Therefore it is really important to know that while the exact distribution is centered at μ , the bootstrap distribution will be centered at x̅

The guidelines to create a bootstrap sample are:

• The bootstrap sample must be the same size as the original sample • The bootstrap sample must contain only values that appear in the original sample - Values from the original sample can appear in the bootstrap sample more than once - Not all values from the original sample need to be in the bootstrap sample

The logic behind establishing confidence levels:

• The level of confidence implies the percent of confidence intervals from all unique random samples that capture the parameter Ex. The idea of a 95% confidence interval is that 95% of those xxx will capture the true population mean

"Rule" for determining the lower and upper bounds of a confidence interval:

• The lower and upper bounds will be the same distance from the point estimate - That is, the point estimate (sample mean, for example) will be halfway between the lower and upper bounds

An executive is concerned with the amount spent on fuel for her company's fleet of cars. She feels that the average miles per gallon of fuel used (MPG) in her fleet of cars is less than 25. Based only on this information, identify the null and alternative hypotheses: 1. The fleet of cars average less than 25 MPG: 2. The fleet of cars average 25 MPG:

1. alternative 2. null

State the appropriate decision for each of the following: reject null hypothesis or fail to reject null hypothesis 1. significance level = 0.05, p-value = 0.001 2. significance level = 0.10, p-value = 0.07 3. significance level = 0.01, p-value = 0.025 4. significance level = 0.05, p-value = 0.075

1. reject null hypothesis 2. reject null hypothesis 3. fail to reject null hypothesis 4. fail to reject null hypothesis

Will scenarios that provide a parameter ever involve inference?

No - Since a parameter is value from a population, no inference is needed in a scenario that provides a parameter as we already know what the value in the population is. Inference is only performed when we don't know the population parameter.

Trial

One repetition of a random experiment

Basic Probability Rules: The probability formula is a _____. The ______ of the fraction cannot be larger than the ______. Both the numerator and denominator are counts which can never be _____.

The probability formula is a fraction. The numerator of the fraction cannot be larger than the denominator. Both the numerator and denominator are counts which can never be negative.

Complement

The probability of an outcome not occurring • 1 − P(outcome)

Power

The probability of correctly rejecting the claim made in the null hypothesis

True or False? A decision involves "rejecting" the claim in the null hypothesis or "failing to reject " the claim in the null hypothesis, but never accepting the null hypothesis as true.

True • We NEVER will accept a null hypothesis as true just as a jury trial NEVER tells a judge the defendant is innocent

True or False? When the population standard deviation is not known, we will use a t-distribution for inference problems instead of a standard normal distribution.

True • When the population standard deviation is not known, we must estimate it with the sample standard deviation. When this happens, we must reference a t-distribution to find things like p-values and critical values for confidence intervals.

True or False? The characteristics of the shape, spread, and center of the sampling distribution of sample means also hold with a bootstrap distribution of sample means.

True - Bootstrap samples are based on the original sample and therefore the bootstrap distribution of bootstrap sample statistics will mimic the characteristics of the sampling distribution of sample statistics.

True or False? The distribution of sample means will always be centered at the population mean.

True - the distribution of sampled means is always centered at the population mean

Which of the following causes an increase in the width of a confidence interval? (Select all that apply) a) An increase in the sample size b) A decrease in the sample size c) An increase in the level of confidence d) A decrease in the level of confidence

b) A decrease in the sample size c) An increase in the level of confidence

Disjoint

Two events are disjoint (or "mutually exclusive") if they both cannot occur in the same replication of a "situation"

A decision is made to reject the null hypothesis. What error could be made? Type I Error or Type II Error

Type I Error

From the histogram and/or summary statistics from the bootstrap distribution of sample means, where is the bootstrap distribution centered? a) the population mean b) the sample standard deviation c) the sample mean d) the population standard deviation

c) the sample mean

20. About 35% of the population has blue eyes based on a study by Dr. P. Sorita Soni at Indiana University. If four different people are randomly selected, what is the probability that they all have blue eyes? a. 0.35 b. 0.65 c. 1.4 d. 0.015 e. 1

d. 0.015

The higher the desired power (probability of correctly rejecting the claim made in the null hypothesis), the _____ the sample size needed. a) Between 0.1 and 0.2 b) The complement of the significance level c) Between 0.8 and 0.9

larger

The population standard deviation describes how the individuals in the population vary. This value will always be _____ than the standard deviation of sample means. You can consider the fact that the mean of a sample will always fall somewhat close to the population mean and will have less spread than individual cases.

larger

In the "significance level" approach to making a decision in a hypothesis test, if the p-value is ____ than the significance level, the decision is to reject the claim in the null hypothesis for the claim in the alternative hypothesis.

less

Probability is the ______ proportion of times an outcome occurs

long-run

Do not attempt to use the formula method to construct a confidence interval for a population _____.

median

If a bootstrap distribution of sample means is heavily skewed, it is better to construct a confidence interval for the _____ rather than the _____.

median, mean

A t-distribution will look like a normal distribution (bell-shaped with one mode), but with ____ area in the tails than a normal distribution. How much more area will depend on the sample size.

_____ and _____ both begin with the letter P, while _____ and _____ both begin with the letter S

parameter and population statistic and sample

The ______ of all cases that have certain values of a variable is the same as the probability that a single case has a certain value

percent

When constructing a confidence interval for a population median, only the ______ method can be used.

percentile

if you are constructing a confidence interval using the ______ method, you will need to generate a bootstrap distribution based on the original data so that it is centered at the sample mean.

percentile

General t-methods formula method to construct a confidence interval for the population mean:

point estimate ± margin of error

A single value that is the best estimate of a population parameter is called ______ estimate, while a range of possible values for a population parameter is called ______ estimate

point, interval

The sample mean will always be _____ in the middle of the confidence interval

right

the best estimate of σ (the population standard deviation) is s, the _____ standard deviation

sample

The best estimate of a population parameter is the _____ ______

sample statistic

A true _____ distribution is a distribution a statistic is calculated for each unique random sample.

sampling

The shape and spread of a bootstrap distribution for a statistic resemble the exact ______ distribution

sampling

The probability the researcher assigns to making a Type I Error is the ______ level in the hypothesis test.

significance • That is, P(Type I Error) = α

A p-value tells us the probability of observing what we did or something more ______ if the null hypothesis is true.

unusual

Statistical inference is concerned primarily with understanding the _____ in our estimates.

variation

A sampling distribution provides us information on how _____ sample statistics are

varied

A point estimate has a _____ _____ level of confidence of being equal to the population parameter. very high or very low?

very low

When randomly selecting people to be part of a sample, we're sampling ______ replacement

without - once a person is selected, they can't be selected again

Categorical data is recorded as a ____ instead of a number.

word

Properties of a the distribution of sample means:

• There is variation in the sample means • The sample means are centered on the population parameter • The distribution of the sample means appeared to be symmetric and even bell-shaped

Percentile

The value such that pth percent of the remaining observations are less than it

Parameter

A number that describes some aspect of a population

Point Estimate

A value of a single statistic that best estimates a parameter in the population Ex. sample mean

Event

An outcome or list of several outcomes for which we want to find the probability of occurring

Bootstrap Distribution

Estimates the sampling distribution; therefore, will allow us to determine a value to use for the standard error of the statistic

True or False? A bootstrap distribution is centered at the population mean.

False

Steps for Obtaining the P-Value:

Step 1: First finds the difference between the null value and the sample mean Step 2: Add this difference to each value in the data set. Step 3: Create an adjusted bootstrap distribution that is consistent with the null hypothesis being true Step 4: Use this adjusted bootstrap distribution to find your p-value

"Capturing a parameter" means:

That the parameter is one of the values listed in the interval

Sample Space

The list of all possible outcomes in an experiment • The probability of, S, the sample space is 1; P(S) = 1

What is the relationship between power and the probability of making a Type II Error (β)?

They are complements Power = 1 - β

The problem with the point estimate:

While the point estimate (i.e. the sample statistic) is the best estimate of a population parameter, it rarely equals the population parameter.

The null hypothesis will always contain an _____ sign

equal

Statistics/Sample notation when you have a single quantitative variable of interest and ONE population:

x̅ = sample mean = the point estimate in a confidence interval s = sample standard deviation SEx̅ = s/√n = standard error of the mean

An economist reported the following. State the type of inference (not an inference problem, estimation, or hypothesis test) for each statement. If the statement did not involve inference, choose "not an inference problem". 1. "Based on comparing a list of 40 common items between last month and this month, prices increased, on average." 2. "Based on a census of all adults between ages 18 and 65, unemployment was 3.8% this month." 3. "Based on a random sample of financial institutions, the average 30-year mortgage rate was 3.125% this month."

1. Hypothesis test - because two groups being compared based on sample of items 2. Not an inference problem - a census means that information is collected on all in the population. Therefore, no inference to the population needs to be made since we have information on the entire population already 3. Estimation - Data were collected from a sample to estimate the average 30-year mortgage rate for this month

1. The null hypothesis is H0: p = 0.2 2. The alternative hypothesis is HA: H0: p > 0.2

Three conditions for one-sample z-methods for a population proportion:

1. The sample is representative of the population of interest 2. The observations in the sample are independent of each other. 3. The distribution of sample proportions is normally distributed. We can assume a normal distribution if the data set contains at least 10 successes and 10 failures.

Once it is time to explore the data, you should obtain summary statistics for the data set, a histogram, and possibly a new graph called Normal Probability Plot. Here's why:

1. The t-statistics will only follow a t-distribution if the sample data come from a population that is normally distributed AND/OR the sample size is sufficiently large 2. To get an idea if the claim in the null hypothesis might be rejected for the claim in the alternative hypothesis

You flip a coin two times. Let event A = both flips are heads Let event B = both flips are tails. 1. Are the two events disjoint? 2. Are the two events independent?

1. The two events are disjoint since the two flips cannot result in both two tails and two heads. 2. The two events are dependent since the probability of event B happening (both tails) would change if event A occurred.

Suppose one card is drawn from a deck of playing cards. Let event A = a club is drawn Let event B = an ace is drawn. 1. Are the two events disjoint? 2. Are the two events independent?

1. The two events are not disjoint since the card drawn can be both a club and an ace (the ace of clubs!) 2. The two events are independent since the probability of the card being an ace will be the same whether the card drawn is a club or any other suit

A family is planning on having three children. Is the family more likely to have all of one sex or two of one sex and one of the other? (We'll assume that it is equally likely that a child born is a girl or boy.) 1. True or False: The outcomes in the sample space are equally likely to occur. 2. What is the probability the family has 2 boys and 1 girl?

1. True - Because we're assuming it is equally likely to have a boy or girl on any one child, all 8 outcomes in the sample space would have the same probability of occurring: .125 2. .375 - From the probability distribution, each outcome in the sample space has a probability of .125. Since the outcomes in the sample space are disjoint, we can add the probabilities of these three outcomes: .125 + .125 + .125 = .375.

P-Value (five points):

1. We start with the belief that the null hypothesis is true 2. The p-value is a probability 3. The p-value is the probability of observing what we did 4. The probability of observing something more unusual or more extreme 5. To help understand what more extreme means, a number line can help

1. Whether or not the person surveyed has a cell phone. 2. Categorical

1. Yes - There are at least 10 successes and 10 failures so we should feel comfortable assuming that the distribution of sample proportions is normally distributed. 2. Yes - Since the batch is large, taking 200 without replacement shouldn't concern us with regard to the independence conditions. Random sampling also helps us feel comfortable that the observations are independent of each other. 3. Yes - A random sample was taken so we should feel comfortable saying the sample is representative of all cups in this large batch.

Udoka wants to know which of her deserts most children at her daughter's school would prefer so she gives her daughter a sample of deserts to give out at lunchtime. Her daughter tallies the votes for each desert type and gives the information back to her mother. 1. Is this an example of inference or not? 2. What type of inference problem is this? Hypothesis Test or Estimation?

1. Yes - Udoka wants to know which of her deserts most children prefer in general, not just those at her daughters school. This means that the population is all children and the sample is those children that voted for the deserts at her daughter's school. Therefore, this is an inference problem. 2. Estimation - Since Udoka is interested in determining which desert children like most she wants to estimate a value that will let her know how many children like each desert.

Research has shown that, for baseball players, a good hip range of motion results in improved performance and decreased body stress. A study was performed on independent samples of 40 professional pitchers and 40 professional position players. The hip range of motion (in degrees) was measured and recorded for all players. 1. What methods should be used? a) two-sample b) paired t-methods 2. Why did you decide on this method? a) The problem describes paired data. b) The problem describes two groups and anytime there are two groups, two-sample t-methods must be used. c) A quantitative variable of interest is measured on all players. d) The problem describes two independent samples were taken. This means that a sample of pitchers and a sample of position players was gathered.

1. a) two-sample 2. d) The problem describes two independent samples were taken. This means that a sample of pitchers and a sample of position players was gathered. • The problem states "A study was performed on independent samples of 40 professional pitchers and 40 professional position players." Since the problem mentions that two independent samples were taken, we can move forward using two-sample t-methods.

Suppose a mobile phone company wants to determine the current percentage of customers aged 50+ who use text messaging on their cell phones. They randomly select 500 of their customers who are 50+ check to see if there is text messaging usage on their accounts. They find 411 customers that use the feature. Have the conditions for one-sample z-methods been met? a. This condition has been met b. This condition has not been met 1. Representative Sample 2. Independent Samples 3. Sampling Distribution Normal

1. a. This condition has been met 2. a. This condition has been met 3. a. This condition has been met

Suppose a consumer group suspects that the proportion of households that have three cell phones is 30%. A cell phone company has reason to believe that the proportion is not 30%. Before they start a big advertising campaign, they conduct a hypothesis test. Their marketing team surveys 150 randomly selected households in their advertising area with the result that 43 of the households have three cell phones. 1. Would you assume that the condition of a representative sample be met? 2. Would you assume that the condition of independent observations will be met? 3. Would you assume that the condition of a normal sampling distribution be met? a. We can assume the condition has been met b. We cannot assume the condition has been met

1. a. We can assume the condition has been met 2. a. We can assume the condition has been met 3. a. We can assume the condition has been met

You plan to conduct a survey on your college campus to learn about the political awareness of students. You want to estimate the true proportion of college students on your campus who voted in the 2012 presidential election. You plan on randomly sampling 15 students from the student directory and contacting them to fill out a survey. 1. Would you assume that the condition of a representative sample be met? 2. Would you assume that the condition of independent observations will be met? 3. Would you assume that the condition of a normal sampling distribution be met? a. We can assume the condition has been met b. We cannot assume the condition has been met

1. a. We can assume the condition has been met 2. a. We can assume the condition has been met 3. b. We cannot assume the condition has been met

A math teacher randomly assigns half his class to do textbook exercises every night for a week. The other half of the class are assigned to complete an equivalent amount of application activities each night. Both the textbook exercises and the application activities cover the same material. At the end of the week, the teacher gives all his students the same assessment to see if one method of instruction results in a higher score, on average, on the assessment. Match each of the following to its component in a hypothesis test: 1. The assessment the teacher gave each student. 2. On average, students who were assigned to complete application activities performed no better on the assessment than those assigned to do the textbook exercises and, in fact, performed slightly worse (on average). 3. The teacher claimed that students assigned to the application activities would perform better on the assessment, on average, than students assigned to do the textbook exercises. a) The hypothesis b) The evidence c) The decision

1. b) The evidence 2. c) The decision 3. a) The hypothesis

Air quality measurements were collected in a random sample of 25 country capitals in 2013 and then again in the same cities in 2014. Researchers would like to use these data to compare average air quality between the two years. 1. What methods should be used? a) two-sample b) paired t-methods 2. Why did you decide on this method? a) The problem describes paired data. b) Paired data occurs anytime there are two groups that are being compared. c) A quantitative variable of interest is measured on all cities. d) Since the cities were first selected randomly, we can consider the data paired data.

1. b) paired t-methods 2. a) The problem describes paired data. • The problem states "Air quality measurements were collected in a random sample of 25 country capitals in 2013 and then again in the same cities in 2014." Since the problem mentions that data was collected on those same cities the following year, we have paired data. We would obtain the difference air quality measurements for all cities (possibly 2013 - 2014 values) and analyze those differences using a paired t-test.

A family is planning on having three children. Is the family more likely to have all of one sex or two of one sex and one of the other? (We'll assume that it is equally likely that a child born is a girl or boy.) 1. Is the family more likely to have all of one sex or two of one sex and one of the other? a) both have the same chance of occurring b) two of one sex and one of the other (such as 2 girls and 1 boy) c) all of one sex (such as all girls) 2. What is the probability that at least one of the children will be a girl?

1. b) two of one sex and one of the other (such as 2 girls and 1 boy) P(2 girls) = P(2 boys) = 3/8, while P(all girls) = P(all boys) = 1/8 2. 1 - ((0.5)(0.5)(0.5) = 1 - .125 = .875

Choose the appropriate adjective for each of the following p-values: a. Strong or convincing b. Some c. Weak d. Not sufficient 1. p-value = 0.0341 2. p-value = 0.075

1. b. Some 2. c. Weak

Marketing companies have collected data implying that teenage girls use more ring tones on their cellular phones than teenage boys do. In one particular study of 20 randomly chosen teenage girls and 20 randomly chosen teenage boys with cellular phones, the mean number of ring tones for the girls was 3.2 with a standard deviation of 1.5. The mean for the boys was 1.7 with a standard deviation of 0.8. Conduct a hypothesis test to determine if the means number of ringtones on girls' phones is greater than for boys. 1. What is the value of the t-statistic? a) -1.7 b) 1.5 c) 3.95 d) 0.9 e) 13 2. What is the degrees of freedom? Use the conservative estimate described in the reading. a) 40 b) 19 c) 20 d) 38

1. c) 3.95 2. b) 19

Summary Statistics set #1 min 96.7 Q1 98 median 98.4 Q3 98.7 max 99.1 mean 98.34 sd 0.5568 n 30 missing 0 Summary Statistics set #2 min 97.95667 Q1 98.27333 median 98.34333 Q3 98.41 max 98.64333 mean 98.34008 sd 0.10134 n 5000 missing 0 1. Summary statistics set #1 are for the? a) bootstrap distribution of sample means b) population data c) sample data 2. Summary statistics set #2 are for the? a) bootstrap distribution of sample means b) population data c) sample data

1. c) sample data 2. a) bootstrap distribution of sample means

Although not given, adjusted bootstrap distributions were formed with the four different sample sizes. Here are the resulting p-values for the different sample sizes. n = 10, p-value = 0.3134 n = 100, p-value = 0.1102 n = 1000, p-value = 0.0003 n = 10000, p-value < 0.0001 For each of these four studies, what decision ( reject null hypothesis or fail to reject the null hypothesis) would be made based on a significance level of 0.05? 1. n = 10 2. n = 100 3. n = 1000 4. n = 10000

1. fail to reject the null hypothesis 2. fail to reject the null hypothesis 3. reject the null hypothesis 4. reject the null hypothesis

Suppose the following bootstrap sample was generated: 17, 22, 6, 17, 22 Calculate and report the bootstrap sample mean for this bootstrap sample to ONE decimal place:

(17 + 22 + 6 + 17 + 22) / 5 = 16.8 months

26. A researcher found a random sample of 35 individuals and recorded the time it took to complete a task. He wished to estimate the mean time it took adults to complete this task. After creating a bootstrap distribution, the following percentiles of the bootstrap distribution are were found and are provided below. Use the percentiles to report a 95% confidence interval for the population parameter. 1% 6.174 2.5% 6.322 5% 6.438 10% 6.593 25% 6.866 50% 7.17 75% 7.481 90% 7.78 95% 7.947 97.5% 8.082 99% 8.304

(6.322 conds, 8.082 seconds)

Calculate the lower and upper bounds of the 95% confidence interval for the mean body temperatures for college students at this particular university. Round to FOUR decimal places. min 97.95667 Q1 98.27333 median 98.34333 Q3 98.41 max 98.64333 mean 98.34008 sd 0.10134 n 5000 missing 0

(98.137ºF, 98.5428ºF) 98.34 - (2)(0.10134) = 98.1373ºF 98.34 + (2)(0.10134) = 98.5427ºF

Formula for a confidence interval for the difference in population means μ1−μ2:

(x̅1 − x̅2) ± (tdf∗)(SEx̅1 − x̅2)

-3.8 = (1.5-1.75)/(.55/√70) = -3.8 x̅ = 1.5, s = .55, n = 70, Ha: µ > 1.75

Consider the following confidence intervals: ($125, $135). (Both of the following answers should be reported as integers.) 1. What is the sample mean amount spent on groceries per week for a family of four? 2. What is the margin of error?

1. $130 2. $5

A fair coin is tossed two times. In a previous question, we determined that the outcomes in the sample space are equally likely to occur. Calculate the following probabilities (report all answers as a decimal to TWO decimal places). 1. P(both tosses result in heads landing face up) 2. P(one toss results in a head and the other toss results in a tails 3. P(both tosses result in tails landing face up)

1. .25 2. .50 3. .25

Return to the problem from the last page. On June 28, 2012 the U.S. Supreme Court upheld the much debated 2010 healthcare law, declaring it constitutional. A Gallup poll released the day after this decision indicates that 466 of 1,012 Americans agree with this decision. These data provide evidence that less than 50% of the U.S. population agrees with the decision (one-sided p-value 0.006). Confidence Level, z* Critical Value: 90% 1.645 95% 1.96 99% 2.576 Construct a 95% confidence interval for the true proportion of Americans that agrees with the decision at the time of the survey. 1. What is the sample proportion? 2. What is the z-critical value needed for this problem? 3. What is the standard error for the problem? 4. What is the lower bound of the confidence interval? 5. What is the upper bound of the confidence interval? 6. Interpret the confidence interval in the context of the problem. Use the pull-down options to complete the sentence. With (90%, 95%, 99%, a lot, some) confidence, the true (proportion, mean) of Americans that agrees with the U.S. Supreme Court decision of declaring the healthcare law constitutional is estimated to be between (0.0157, 0.46, 0.429, 0.491) and (0.0157, 0.46, 0.429, 0.491), with a point estimate of (0.0157, 0.46, 0.429, 0.491). g

1. .46 = 466/1012 2. 1.96 3. .0157 = √(.50(1−.50)/1012) 4. .429 = .46 - (1.96*.0157) 5. .491 = .46 + (1.96*.0157) 6. With (95%) confidence, the true (proportion) of Americans that agrees with the U.S. Supreme Court decision of declaring the healthcare law constitutional is estimated to be between (0.429) and (0.491), with a point estimate of (0.46).

Suppose a faculty member wanted to know if students actually spend the appropriate amount of time outside of class studying, as she feels students are not spending enough (8-12 hours) time outside of class on her course. There are two competing hypotheses in this problem. Which is the null hypothesis and which is the alternative hypothesis? 1. Students spend the recommended 8 hours per week, on average, doing work outside of this 4-credit class is the _____ hypothesis. 2. Students spend less than the recommended 8 hours per week, on average, doing schoolwork outside of this 4-credit class is the _____ hypothesis.

1. null • This claim is something we may start with if we don't know any better 2. alternative • This is the claim the researcher believes might be happening. Remember, the alternative hypothesis generally goes with the claim the researcher is making or what the researcher is wondering.

Suppose we flip a coin 10 times and obtain 8 heads. 1. What is a trial in this random experiment? 2. How many trials are there in this random experiment? 3. What is the outcome on each trial?

1. one flip of the coin 2. 10 3. the result of each flip of the coin (head or tails)

Jeremy wanted to estimate the mean diameter of a certain type of tree in an old growth strand of trees. He took a random sample of 25 trees and calculated a mean of these 25 trees of 8.8 feet. He also reported to his supervisor a range of values he thought the mean diameter of all trees in this old growth area could be: 8 feet to 9.6 feet. Which is the point estimate and which is the interval estimate? 1. 8.8 feet is the? 2. 8 to 9.6 feet is the?

1. point estimate 2. interval estimate

Suppose α = 0.05. What decision would be made for the following p-values? reject null hypothesis or fail to reject null hypothesis 1. p-value = 0.0001 2. p-value = 0.0499 3. p-value = 0.0501

1. reject null hypothesis 2. reject null hypothesis 3. fail to reject null hypothesis

For each of the following scenarios, determine if t-methods could be used for inference: 1. Sample size of 10, population is thought to be heavily skewed, the population standard deviation is unknown 2. Sample size of 100, population is thought to be slightly skewed, the population standard deviation is unknown

1. t-methods CANNOT be used 2. t-methods can be used • If the sample size is large enough or the population is normal, the distribution of the sample mean will be approximately normal and that is when the t-methods can be used for inference

For each of the following scenarios, determine if t-methods could be used for inference: 1. Sample size of 10, the population is normal and the population standard deviation is unknown 2. Sample size is 100, the population is normal and the population standard deviation is unknown

1. t-methods can be used 2. t-methods can be used • If the sample size is large enough or the population is normal, the distribution of the sample mean will be approximately normal and that is when the t-methods can be used for inference

H0: p = 0.5 HA: p ≠ 0.5 90% confidence interval for p: (0.48, 0.74) At what significance level is the above decision made? (Report a numerical value only as a percentage.) ___ %

10% = 100-90

H0: μ = 10 HA: μ ≠ 10 90% confidence interval for μ: (15, 20) At what significance level is the above decision made? (Report a numerical value only as a percentage.) ___ %

10% = 100-90

A biologist is interested in studying the effects of applying insecticide to a fruit farm on the local bat population. She collects a random sample of 30 bats from each of the 5 major fruit farms in this area (150 bats total) and finds a mean weight from this sample to be 150.4 grams. Previous research suggests that bats in this area should have a mean weight an average of 157 grams. 10. What type of study was being conducted? a. A simple random study b. An experiment with 5 levels to the explanatory variable (the 5 different farms) c. An experiment with only one level (the insecticide) d. An observational study 11. If previous research suggests that bats in this area should have a mean weight of 157 grams, what is the null hypothesis for this test? a. H0: μ = 157 b. H0: μ < 157 c. H0: μ = 150.4 d. H0: μ < 150.5 e. H0: μ = 24.6

10. d. An observational study 11. a. H0: μ = 157

One hundred eight Americans were surveyed to determine the number of hours they spend watching television each month. A 95% confidence interval was found to be (142 hours, 163 hours). The confidence interval was calculated using t-methods. What is the margin of error?

10.5 hours • The range of the confidence interval as 21 hours, which include two margins of error. You have 21 hours/2 = 10.5 hours

Sam surveys 150 adults who he believes are representative of all adults in a particular community. One hundred twenty say they are in favor of a certain issue on the ballot. Every adult in the sample has only two choices: favors the issue or does not favor the issue. Calculate the probability that a randomly selected person in this population will be in favor of this issue? Report your answer to ONE decimal place.

120 / 150 = .8

The U.S. Census Bureau reports that the average number of employees in a small business is 16.1. A random sample 45 small businesses are contacted and were asked how many employees were on their payroll. A sample mean of 17.6 employees was found. 13. What is the mean of the distribution of the sample mean? a. 45 b. 16.1 c. 17.6 d. 5 e. 0.7454 14. If the population standard deviation was known to be 5, according to the Central Limit Theorem what is the standard deviation of the distribution of the sample mean? a. 45 b. 16.1 c. 17.6 d. 5 e. 0.7454

13. b. 16.1 14. e. 0.7454

On the previous page, the 95% confidence interval for p (the proportion of all adults in 2005 who chose math as their favorite subject) from my simulation was (0.204, 0.256). The correct interpretation of this interval is, "We're 95% confident that between 20.4% and 25.6% of all adults in 2005 chose math as their favorite subject while in school." From my simulation above, a 99% confidence interval for p is (0.197, 0.265). State if each of the following is true or false regarding how the interpretation of the 99% confidence interval would change from the interpretation of the 95% confidence interval? 1. The interpretations would be exactly the same - nothing would change. 2. The level of confidence would change from 95% to 99% 3. The percentages would change to 19.7% and 26.5%, respectively. 4. The population would change. 5. Since the level of confidence is 99%, we could remove "We're 95% confident" and just say, "We're almost certain".

1. False 2. True 3. True 4. False 5. False

152.5 hours • If you took the lower bound, 142 hours, and added the value of the margin of error, 10.5 hours, the result is the point estimate, 152.5 hours.

What is the z-score used in the construction of a 95% confidence interval?

2 - Since the sample means are approximately normally distributed, 95% of the bootstrap sample means will fall within two standard deviations of the original sample mean. A z-score of 2 comes from the 68-95-99.7 rule.

What is the hypotheses notation for each? 1. Students spend the recommended 12 hours per week, on average, doing work outside of this 4-credit class is the null hypothesis. 2. Students spend less than the recommended 12 hours per week, on average, doing schoolwork outside of this 4-credit class is the alternative hypothesis.

1. H0 : μ = 12 hours per week • The null hypothesis is consistent with the idea of no difference. The amount of time students are spending outside of class on schoolwork is no different than the recommended 8 hours. 2. HA: μ < 12 hours per week • The alternative hypothesis is associated with the claim the faculty member is making prior to collecting any data - students in her class, on average, are not spending the appropriate amount of time outside of class on her course.

Calculate the t-statistic for the following information. Round to 2 decimal places. x̅ = 15, s = 5.3, n = 35, Ha: µ > 13

2.23 = (15-13)/(5.3/√35)

What percent of the bootstrap statistics would be less than the lower bound of a 95% confidence interval constructed using the formula method?

2.5%

1. H0: p = 0.2 2. HA: p ≠ 0.2

On June 28, 2012, the U.S. Supreme Court upheld the much debated 2010 healthcare law, declaring it constitutional. A Gallup poll released the day after this decision indicates that 466 of 1,012 Americans agree with this decision. Does this provide evidence that less than 50% of the U.S. population agrees with the decision? 1. What is the alternative hypothesis? 2. What is the null hypothesis? 3. What is the sample proportion? 4. What is the z-statistic? 5. What is the conclusion in the context of the problem. Fill in the blank We (accept, fail) the null hypothesis with a p-value of (-2.54, 0.006, 0.46, 0.05) at the (-2.54, 0.006, 0.46, 0.05) significance level. The data provide (no, moderate, weak, strong) evidence to suggest that the proportion of Americans that agree with the decision at the time of the survey is (less than, greater than, equal to, different than) 50%.

1. Ha: p < 0.5 2. Ho: p = 0.5 3. .46 = 466/1012 4. -2.54 = (.46−.50) / (√(.50(1−.50)/1012) 5. We (fail) the null hypothesis with a p-value of (0.006) at the (0.05) significance level. The data provide (strong) evidence to suggest that the proportion of Americans that agree with the decision at the time of the survey is (less than) 50%.

Oregon State University is interested in determining the average amount of paper, in sheets, that is recycled each month. In previous years, the average number of sheets recycled per bin was 59.3 sheets, but they believe this number may have increase with the greater awareness of recycling around campus. They count through 79 randomly selected bins from the many recycle paper bins that are emptied every month and find that the average number of sheets of paper in the bins is 62.4 sheets. They also find that the standard deviation of their sample is 9.86 sheets. What is the value of the test-statistic for this scenario?

2.794 = (62.4-59.3)/(9.86/√79) = 2.794

If the data in the population are slightly skewed, typically a sample size of __ or more is needed for the sample means to be approximately normal

Suppose a statistics instructor believes that there is a significant difference between the mean class scores of the AM statistics students and PM statistics students on Exam 2. She takes random samples from each of the populations. She finds out that the average score of AM students is 5 points higher than PM students. She finds, with 95% confidence, the mean score for the AM students is between 3 to 7 points higher than the mean for the PM students. 5 points is a (point estimate or confidence interval) while 3 to 7 points is a (point estimate or confidence interval)

5 points is a (point estimate) while 3 to 7 points is a (confidence interval)

5% = 100-95 • When using a confidence interval to make a decision about a two-sided hypothesis test, the significance level will always be the complement of the confidence level.

Bootstrapping

Any test or metric that relies on random sampling with replacement - Can be used to estimate population standard deviation (σx̅ = σn/√σx) - Allows assigning measures of accuracy (in terms of bias, variance, confidence intervals, prediction error, etc.) to sample estimates

Degrees of Freedom

Are the number of data points that are free to change when a statistic (or parameter) is fixed. In other words, it is the number of independent pieces of information you have to work with Ex. If a student needs a 90 average based on 3 exams (fixed x̅ = 90) and the first two test scores are 82 and 95. That means the last test must be 93 in order to get an average of 90. Since the first two scores could have been anything, there are 2 degrees of freedom here.

A political science professor at a large university wants to estimate the percent of students who are registered voters. He surveys 500 students and finds that 300 are registered voters. If the variable of interest is whether or not a student surveyed is registered to vote, this variable is Quantitative or Categorical?

Categorical

Is this quantitative or categorical? Investors in the stock market are interested in the true proportion of stocks that go up and down each week. Businesses that sell personal computers are interested in the proportion of households in the United States that own personal computers. Confidence intervals can be calculated for the true proportion of stocks that go up or down each week and for the true proportion of households in the United States that own personal computers.

Categorical • Researchers had to determine if each selected household had a personal computer and then recorded either "yes" or "no". In each example, we have categorical data.

Is this quantitative or categorical? During an election year, we see articles in the newspaper that state confidence intervals in terms of proportions or percentages. For example, a poll for a particular candidate running for president might show that the candidate has 40% of the vote within three percentage points. Often, election polls are calculated with 95% confidence, so, the pollsters would be 95% confident that the true proportion of voters who favored the candidate would be between 0.37 and 0.43: (0.40 - 0.03 and 0.40 + 0.03).

Categorical • They probably asked the question, "Who do you intend to vote for, Candidate A or B?" and then recorded the responses. Each response was a word.

Effect Size

A quantitative measure of the strength of a phenomenon

Bootstrap Sample

A random sample from the original sample with replacement, using the same size as the original sample

Bootstrap Statistic

A statistic, such as a mean or a proportion, that is calculated from a bootstrap sample - If the variable is quantitative, the bootstrap statistic might be the mean, median, or difference in means or medians - If the variable is categorical, the bootstrap statistic might be the proportion or difference in proportions.

The Oregon Department of Transportation is interested in the average number of miles driven by working adults aged 35 - 46 in a one week period. ODOT believes that working adults in this age range drive more than 100 miles in a one week period. ODOT sends a survey to 1000 such adults and surveys are returned by 274 adults. The survey asks respondents for an estimate of how many hours they believe they drive each week. ODOT determines that the average number of miles driven each week by the respondents of the survey is 74.9 miles. The investigators at ODOT generated 10,000 adjusted bootstrap samples and calculated the mean of each. Below is a table showing how many adjusted bootstrap sample means are in each region. ≤ 74.9 miles: 154 between 74.9 and 100 miles: 4862 ≥ 100 miles: 4984 Using this table, calculate and report the p-value to FOUR decimal places.

.0154 • To calculate the p-value we need to determine how many adjusted bootstrap sample means are as or more unusual than what we observed in the sample. From the previous question, any sample mean ≤ 74.9 is considered as or more unusual if the null hypothesis is true. 154 of the 10,000 adjusted bootstrap sample means are "as or more unusual", giving us a p-value of 154/10000 = 0.0154.

.0306 - P(A) = 13/52, P(B) = 12/51 (given event A occurred), P(C) = 26/50 (given events A and B both occurred) Applying the general multiplication rule, P(A and B and C) = (13/52)(12/51)(26/50)

.25 was*.25*.5 = .0313

.521 = (.09-.08) / √(.08(1−.08)/200)

Suppose a 99% confidence interval for a parameter is constructed. At what percentile are the lower and upper bounds?

.5th and 99.5th percentiles

An unbalanced coin is tossed two times. On this coin, the probability of getting a heads to land face up on any one toss is 0.4. There are four possible outcomes in the sample space (HH, HT, TH, TT), where HT (for example) means heads on first toss AND tails on second toss. If the outcomes in the sample space are disjoint, use the addition rule to find the probability of getting at least one head on the two tosses. Report your answer to TWO decimal places.

.64 = P(HT) + P(TH) + P(HH) = 0.24 + 0.24 + 0.16

The probability of an outcome is always between:

0 and 1, inclusive • That is, 0 ≤ P (outcome) ≤ 1

Typical probabilities of making a Type II Error are between ___ and ___.

0.1 and 0.2

A probability closer to ___ indicates a higher likelihood of occurring.

If an outcome will definitely occur, what is P(outcome)?

The sum of the probabilities of all possible outcomes in the sample space is?

A family is planning on having three children. Is the family more likely to have all of one sex or two of one sex and one of the other? (We'll assume that it is equally likely that a child born is a girl or boy.) 1. What is the probability that all three children are boys? 2. What is the probability that the first born is a boy and the next two born are girls?

1 .125 = .5*.5*.5 2. .125 = .5*.5*.5

H0: μ1 = μ2 HA: μ1 ≠ μ2 99% confidence interval for μ1 - μ2: (-0.25, 1.43) At what significance level is the above decision made? (Report a numerical value only as a percentage.) ___ %

1% = 100-99

Because researchers want to keep the probability of making such an error low, typical significance levels the researcher may use are __%, __%, or __%

1%, 5%, or 10%

Based on the p-value, we will make one of two decisions:

1) Reject the claim in the null hypothesis and say the claim in the alternative hypothesis is true OR 2) Do not reject the claim in the null hypothesis. That is, decide that there is not enough evidence to conclude the claim in the alternative hypothesis is true.

Consider the random experiment of rolling a fair six-sided die once. What is the sample space of this random experiment?

1, 2, 3, 4, 5, 6

Data were collected in North Carolina over the period of a year on 150 randomly selected mothers and their newborn babies. Although many variables were measured, researchers were interested in whether or not the mother was a smoker and the weight of their newborn baby. Researchers would like to know if there was evidence that newborns from mothers who smoke have a different average birth weight than newborns from mothers who don't smoke. The following summary statistics were found on each group of newborns. Consider the samples of newborns as representative of all newborns born to smoking and nonsmoking mothers. Smoker mean 6.78 standard deviation 1.43 sample size 50 Nonsmoker mean 7.18 standard deviation 1.6 sample size 100 Do the data provide evidence of a difference in birth weight for babies born to mothers who smoke and mothers who do not? Use a significance level = 0.05. 1. What is the value of the test statistic? Be sure to use the difference as μnon−smoking − μsmoking 2. What is the degrees of freedom? Use the conservative estimate recommended in the reading

1. 1.551 (7.18-6.78) sqrt(1.60^2)/(100) +(1.43^2)/(50) 2. 49 = smaller sample size minus 1

1. 71,492 - the sample mean (x̅) 2. 2 - z-score of 2 can be used in the construction of a 95% confidence interval 3. 8.8 - margin of error = (z-score)(SEx̅) = 2*4.4 4. ( 71,483.2 km, 71,500.8 km) - 1,492 - 8.8 = 71,483.2 km, 71,492 + 8.8 = 71,500.8 km

A family is planning on having three children. Is the family more likely to have all of one sex or two of one sex and one of the other? (We'll assume that it is equally likely that a child born is a girl or boy.) 1. How many outcomes are in the sample space? 2. Is the sex of newborn children independent? a) No. If the couple's first child is a boy (for example), the second child is more likely to be a girl. b) Yes since the sex of one child in the family does not depend on the sex of any child born to this couple previously. c) No. Genetics may play a role making it more likely that a child will be one particular sex than the other.

1. 8 - The possible outcomes are BBB, BBG, BGB, GBB, GGB, GBG, BGG, GGG 2. b) Yes since the sex of one child in the family does not depend on the sex of any child born to this couple previously.

Sampla Data: min 96.7 Q1 98 median 98.4 Q3 98.7 max 99.1 mean 98.34 sd 0.5568 n 30 missing 0 Bootstrap Distribution of Sample Means: min 97.95667 Q1 98.27333 median 98.34333 Q3 98.41 max 98.64333 mean 98.34008 sd 0.10134 n 5000 missing 0 1. What is the value of x̅ that will be used in the construction of a 95% confidence interval? 2. What is the value of SEx̅ that will be used in the construction of a 95% confidence interval?

1. 98.34 2. 0.10134

1. A Type __ error occurs when a decision is made to reject the claim made in the null hypothesis and in fact the null hypothesis is actually true. 2. A Type __ error occurs when a decision is made not to reject the claim in the null hypothesis and in fact the null hypothesis is actually false and should have been rejected.

1. A Type I error 2. A Type II error

Facts about the distribution of the sample proportion: If a sample proportion is calculated from a random sample of size n from a population with proportion p, then the sampling distribution of p^ has the following properties:

1. A shape that is approximately normal as long as the sample size is considered sufficiently large. The sample size needed to ensure a normal shape for the sampling distribution will depend on the value of the population proportion. We can assume a normal distribution if the data set contains at least 10 successes and 10 failures. 2. At the center of this sampling distribution will be the population proportion p. In other words, the mean of all possible p^ is equal to the population proportion. Using notation we can say μp^ = p 3. The standard deviation for the distribution of p^ equals σp^ = √p(1−p) . One issue here is that we typically do not know the exact value for p, so what value we used in this formal will change slightly depending on if we are performing a hypothesis test or calculating a confidence interval.

1. Condition has been met 2. Condition has been met 3. Condition has been met 4. Condition has been met

Three Points on Distribution of Sample Means:

1. Every random sample produces a slightly different sample mean. Therefore we say there is variation in the sample means. 2. You may have noticed that many of the sample means are near the population mean, but not all are equal to this value. It appeared that the distribution of sample means is centered at the population mean. 3. When plotting all of the sample means, it looks like they form a symmetric, normal shaped distribution when the population started out as a normal shape or when the sample size was large. Will this always be true?

Below you will find the steps of conducting a hypothesis test using t-methods. Put the steps in order from first to last. • Calculate the t-statistic • State the conclusion of the test in the context of the problem • Find degrees of freedom • Explore the sample; be able to determine if normally distributed • Find the p-value using the appropriate t-distribution

1. Explore the sample; be able to determine if normally distributed 2. Calculate the t-statistic 3. Find degrees of freedom 4. Find the p-value using the appropriate t-distribution 5. State the conclusion of the test in the context of the problem

The Berkman Center for Internet & Society at Harvard recently conducted a study analyzing the privacy management habits of teen internet users. Researchers when into two classrooms in a local high school classroom and asked each student if they had more than 500 Facebook friends. Of the 50 teens surveyed, 7 reported having more than 500 friends on Facebook. 1. Has the representative sample condition been met? 2. Has the independence condition been met? 3. Has the normal sampling distribution been met?

1. No - The problem says that two classrooms in a local high school were sampled. We may be concerned that the responses from students in these two classrooms are not representative of all teens. 2. No - The problem says that two classrooms in a local high school were sampled. It seems reasonable to argue that students in these classrooms may all know each other and may even be friends on Facebook. It doesn't appear that the number of friends one student has would be totally independent of the number of Facebook friends another student in the classroom has. 3. Yes - From the sample of 50, we are told that only 7 students have more than 500 friends. This is less than 10 successes and, therefore, we cannot assume that the distribution of sample means will be normally distributed.

The Two Competing Hypotheses in a Hypothesis Test Problem:

1. Null Hypothesis 2. Alternative Hypothesis

1. Null Hypothesis: H0: p (=) (0.08) 2. Alternative Hypothesis: HA: p (>) (0.08)

For each scenario below, state if the given value is a statistic or parameter. 1. A fifth grade teacher wants to know what percent of his students read at least 30 minutes per night. He asks each student in his class if they read at least 30 minutes per night. 40% said they did. Is 40% a parameter or statistic? 2. An energy official wants to estimate the average oil output per well in the United States. From a random sample of 40 wells throughout the United States, the official obtains a mean of 10.1 barrels per day. Is 10.1 a parameter or statistic? 3. 40% of the 150 workers at a particular factory were paid less than $40,000 per year. You have the payroll data for all of the workers. Is 40% a parameter or statistic?

1. Parameter 2. Statistic 3. Parameter

Which of the following scenarios would result in a sampling distributing with an approximately normal shape? 1. Slightly skewed population, sample size of 10 2. Slightly skewed population, sample size of 100

1. Sampling distribution NOT normal 2. Sampling distribution normal • When the population is normal, the sampling distribution will also be normal no matter what the sample size is • With a skewed population and a small sample size, it is likely that the distribution of sample means will also be skewed

Which of the following scenarios would result in a sampling distributing with an approximately normal shape? 1. Normal population, sample size of 10 2. Normal population, sample size of 50

1. Sampling distribution normal 2. Sampling distribution normal • When the population is normal, the sampling distribution will also be normal no matter what the sample size is • With a skewed population and a small sample size, it is likely that the distribution of sample means will also be skewed

For each scenario below, state if the given value is a statistic or parameter. 1. A researcher wants to estimate the average height of men aged 20 years or older. From a simple random sample of 40 women, the researcher obtains a mean height of 70.1 inches. Is 70.1 inches a parameter or statistic? 2. 85% of the 100 United States senators voted for a particular measure. Is 85% a parameter or statistic?

1. Statistic 2. Parameter

For each scenario below, state if the given value is a statistic or parameter. 1. It is recommended that children ages 2 to 18 consume fewer than 6 teaspoons of sugar (about 25 grams) each day. Suppose a nutritionist wanted to determine if the mean amount of sugar consumed by children aged 2 to 18 is indeed less than 6 teaspoons a day. From a random sample of 150 children aged 2 to 18, the nutritionist obtains a mean of 19 teaspoons of sugar consumed each day. Is 19 teaspoons per day a parameter or statistic? 2. 30% of a random sample of over 1000 dog owners poop scoop after their dog. Is 30% a parameter or statistic?

1. Statistic 2. Statistic

Complete the following table by selecting whether or not the t-methods can be used or should not be used for the given scenario: 1. The sample size is 25 and the sample data are heavily right skewed and bi-modal with many outliers. 2. The sample size is less than 30 and the sample data are symmetric and bell shaped with the population standard deviation known. 3. The sample size is greater than 30 and the sample data are symmetric and bell shaped.

1. T-methods should not be used 2. T-methods can be used 3. T-methods can be used

The Three Components of a Hypothesis Test:

1. The Hypotheses 2. The Evidence 3. The Decision

Three Probability Rules

1. The Multiplication Rule 2. The Addition Rule 3. The Complement Rule

Two things that affect the width of a confidence interval:

1. The confidence level 2. The sample size

Why is a sampling distribution important? (Two Reasons):

1. The distribution of sample means and confidence intervals 2. The distribution of sample means and hypothesis testing

Suppose that a market research firm is hired to estimate the percent of adults living in a large city who have cell phones. Five hundred randomly selected adult residents in this city are surveyed to determine whether they have cell phones. Of the 500 people surveyed, 421 responded yes - they own cell phones. As a follow-up to their initial question of interest, researchers asked each of the 421 people who reported having a cell phone how many minutes a typical cell phone conversation lasts. The sample yielded a mean of 1.3 minutes and a standard deviation of 0.6 minutes. 1. The variable of interest is? 2. This variable is Quantitative or Categorical?

1. The length of time for a typical cell phone conversation. 2. Quantitative

One university wanted to find out how often students living on the campus use illegal drugs. For each of the 5 dorm buildings owned for the university, 10 randomly selected dorm rooms were visited. A uniformed police officer interviewed the student living in the dorm room selected. The officer asked the name of the participant and if he or she has used an illegal drug in the last month. No one in the study admitted to using illegal drugs, therefore the school concluded it was truly a drug free campus. 5. What type of sampling design was used? a. Convenience sampling b. Simple random sampling c. Observational sampling d. Stratified random sampling e. Cluster sampling 6. Which of the statements below is true about the conclusion made in this study? a. Respondents may not answer truthfully so these results may contain some response bias. b. Since a uniform police officer was conducting the interview, we can guarantee that participants were being honest. c. We must know the size of the campus to determine if these results would be valid for the whole study body. It is unclear if the sample size is a large enough. d. There is absolutely nothing wrong with this conclusion.

5. d. Stratified random sampling 6. a. Respondents may not answer truthfully so these results may contain some response bias.

Suppose a 90% confidence interval for a parameter is constructed. At what percentile are the lower and upper bounds?

5th and 95th percentiles

69 • Degrees of freedom are n - 1 so we have 70 - 1 = 69

Consider the following process: A person flips a coin to determine if a balanced six sided die will be rolled. If the coin comes up heads the die will be rolled once and a number will be observed. If the coin comes up tails the die is not rolled and the experiment ends. How many possible outcomes are there for this random process?

Oregon State University is interested in determining the average amount of paper, in sheets, that is recycled each month. In previous years, the average number of sheets recycled per bin was 59.3 sheets, but they believe this number may have increase with the greater awareness of recycling around campus. They count through 79 randomly selected bins from the many recycle paper bins that are emptied every month and find that the average number of sheets of paper in the bins is 62.4 sheets. They also find that the standard deviation of their sample is 9.86 sheets. In the last problem you found the t-statistic. What are the degrees of freedom for this t-test?

78 • The degrees of freedom are n - 1. In this example we have 79 - 1 = 78

Probability of making a Type II Error points:

A Type II Error occurs when the decision is made NOT to reject the claim in the null hypothesis when in fact the claim in the null hypothesis is false. • The notation for the probability of making a Type II Error is β (the Greek letter beta). • P(Type II Error) = β • There is not an easy way to calculate β. • In fact, there is no single value for β! That's because there are many possible values the population parameter could be to make the null hypothesis false.

Alternative Hypothesis

A competing hypothesis that a researcher often claims as true • The alternative hypothesis is consistent with idea the researcher is hoping to prove or support

Which form of technology would be best to use for simulating a random experiment?

A computer

Type II Error

A decision is made not to reject the claim in the null hypothesis when in fact the claim made in the null hypothesis is not true

Type I Error

A decision is made to reject the claim made in the null hypothesis when the claim in the null hypothesis is in fact true

One-Sided Hypothesis Test

A hypothesis test is performed when the alternative hypothesis is a < (less than) or < (greater than) • For one-sided hypothesis tests, only consider one side of the number line for values that are considered "more unusual".

Two-Sided Hypothesis Test

A hypothesis test is performed when the alternative hypothesis is ≠ (not equal to) • For two-sided hypothesis tests, consider both sides of the number line for values that are considered "more unusual"

Confidence Interval

A list (or "range") of possible values a researcher believes a population can be with a certain degree of confidence

Probability Distribution

A list (usually in table format) of all the possible outcomes and their probabilities Ex. outcome probability 2 heads .25 2 tails .25 1 head, 1 tail .50

Sampling Distribution

A list of all possible values of the sample statistic calculated from random samples of the same size from the same population • Could be the mean, proportion, difference in sample means, difference in sample proportions, or any other statistic you can think of • Often called the distribution of sample means

Standard Deviation

A measure of spread • You have to be thoughtful of what type of distribution you are referencing as you decide what standard deviation to use.

Statistic

A number that is computed from data in a sample

Level of Confidence

A numerical value of how confident that the population parameter is one of the values in the interval estimate; level of certainty

Significance Level

A predetermined value such that any p-value less than or equal to the significance level will lead to the decision to reject the claim in the null hypothesis for the claim in the alternative hypothesis.

P-Value

A probability of observing a statistic from a study like the one observed or "more unusual" than the one observed if the null hypothesis is true • The probability of observing what we observed in the study. But, also included in the p-value is the probability of observing something more unusual, or sometimes stated more extreme, than what was observed in the study.

Random Experiment

A process by which we observe something uncertain

P-Value

The probability that a random sample will produce a sample mean like the one observed, or something more extreme

You are part of a jury that makes a decision that the defendant is guilty. This means that the jury believed the probability of observing the evidence if the defendant was innocent was a) very high (maybe around 0.99) b) around 0.5 c) very low (maybe around 0.01)

c) very low (maybe around 0.01)

True or False? If the population data are slightly skewed, a sample size of 30 or more is typically large enough for the distribution of sample means to be approximately normal.

True - while 30 is typically a large enough sample size for the sample means to be approximately normal, it may have to be much larger if the population data are extremely skewed

Lady Tasting Tea Example: The first component of a hypothesis test is to determine the hypotheses. There are two competing hypotheses in this example: 1. The lady is just guessing 2. The lady is not guessing and can really tell the difference in the taste of the tea between the two preparations

In this problem, the claim the lady is making is that she can distinguish the taste of tea between tea added to milk and tea where milk is added to tea. Therefore, choice 2) above would be the alternative hypothesis. The competing hypothesis is the null hypothesis. The idea of no effect or no difference in this problem is that she cannot tell the difference in the taste of the tea, or that the preparation of the tea has no effect on the taste of the tea. Therefore, choice 1) above is the null hypothesis. H0: the lady is just guessing HA: the lady is not guessing If she was just guessing, what proportion of the cups would she correctly identify the preparation method would be 0.5. In statistical notation, the null hypothesis would be: p = 0.5 Therefore, the hypotheses could be written this way: H0: p = 0.5, which implies the lady is just guessing HA: p > 0.5, which implies the lady is not guessing and can really tell the difference in the taste of tea between the two preparations. where p = the proportion of cups the lady correctly identifies the preparation method.

Estimation Problem

Involves estimating the value of the parameter of interest based on the information we gather from the sample Ex. Our sample mean is the best guess as to the value of the population mean

Why wouldn't a researcher always set the significance level at a very low level like 0.01?

It's because there is a trade-off - with a very low significance level, a researcher would need a really, really small p-value in order to reject the null hypothesis. This will set them up for making a "fail to reject the null hypothesis" conclusion more often which may be the wrong conclusion is some cases. That is why a significance level of 0.05 is typically used.

Probability

Long-run proportion (or relative frequency) of times the outcome occurs when a random experiment is performed under identical conditions

A fifth grade teacher wants to know what percent of his students read at least 30 minutes per night. He asks each student in his class if they read at least 30 minutes per night. 40% said they did. Is this an inference problem?

No - The population is all students in this teacher's fifth-grade class. Since the teacher collected information on all in his population of interest, no inference is needed.

Two balanced six-sided dice are rolled at the same time (a red die and a green die). Let A = {doubles are rolled}. Calculate and report P(A). (Recall, A = doubles are rolled). Round your answer to FOUR decimal places.

Notice that 6 of the 36 outcomes result in doubles (1,1), (2,2), (3,3), (4,4), (5,5), (6,6). The equally likely formula can be used since each of the 36 outcomes in the sample space are equally likely to occur since both dice are "balanced". Therefore, P(A) = 6/36 = 1/6 = .1667

A student wondered if more than 25% of college students own a pet. Compose the null and alternative hypotheses in notation. H0: HA:

Null: H0: p = .25 Alternate: HA: p >.25

The owners of a small gardening company are interested in the average time it takes a rose in Portland to reach full bloom so that they can determine how early they should begin taking orders for their rose selling business. They petition 400 randomly selected residents of Portland to watch the roses in their garden and report how long (in weeks) it takes for each rose in their rosebushes to reach full bloom. They believe that the average time it takes a rose in Portland to reach full bloom is less than 8 weeks. The null hypothesis will have the form: The alternative hypothesis will have the form:

Null: H0: μ = 8 weeks Alternate: HA: μ < 8 weeks

An unbalanced coin is tossed two times. On this coin, the probability of getting a heads to land face up on any one toss is 0.4. There are four possible outcomes in the sample space (HH, HT, TH, TT), where HT (for example) means heads on first toss AND tails on second toss. Find the probability distribution (i.e. probability of each of these outcomes occurring). Report each to TWO decimal places. P(HH) P(TT) P(HT) P(TH)

P(HH) = .16 P(TT) = .36 P(HT) = .24 P(TH) = .24

Probability Formula:

P(outcome) = number of times the outcome occurs / total number of trials

Conditional Probabilities

Probabilities calculated based on whether or not a certain event already occurred

Interval

Provides a range of possible values for something Ex. A weather forecaster might predict the average high temperature for today between 70 and 75 degrees

Hypothesis Test Problem

Questions of interest that make a claim about the population will involve doing a hypothesis test to test that claim. Often, the claim may be what a researcher wonders or believes to be true in the population. These claims will either: • mention or make reference to a particular value for the population parameter • question if there is a difference in the parameter being measured between two populations.

Outcome

Refers to the result of, or what happens on, a trial

Number Line

Represents the distribution of the sample statistic if the null hypothesis is true • On that number line we will place the null hypothesized value as well as the statistic we observed

Standard Error of the distribution of sample means - formula used for t-methods:

SEx̅ = s/√n

Two Sample t-Statistic degrees of freedom is called what?

Satterthwaite Degrees of Freedom • For the purposes of hand calculations in class, we will use the conservative estimate of smaller sample size minus 1.

Suppose a baker claims that the average bread height is more than 15 cm. Several of this customers do not believe him. To persuade his customers that he is right, the baker decides to do a hypothesis test. He bakes 10 loaves of bread. The mean height of the sample loaves is 17 cm with a sample standard deviation of 1.9 cm. The heights of all bread loaves are assumed to be normally distributed. Step 1: What is the variable of interest? Step 2: For the scenario, you can assume that the 10 loaves are representative of all loaves made by this baker. Step 3: Since the baker wants to find evidence that his loaves rise to more than 15 cm, he will conduct a hypothesis test. Step 4: What is the null hypothesis? H0: What is the alternative hypothesis? Ha: Step 5: The sample appeared to look normal, therefore you can assume that it came from a normal population. It's safe to assume that the distribution of samples is normal by the Central Limit Theorem. Step 6: What is the value of the test statistic? What is the degrees of freedom? Step 7: What is the conclusion of the study in the context of the problem?

Step 1: The height of the loaf made by the baker Step 4: What is the null hypothesis? H0: μ = 15 cm What is the alternative hypothesis? Ha: μ > 15 cm Step 6: What is the value of the test statistic? 3.329 What is the degrees of freedom? 9 Step 7: There is strong evidence to suggest that the mean height of bread loaves made by this baker is greater than 15 cm.

The 5 beginning steps of inference:

One-Sample Z-Test Steps for a Population Proportion:

Step 1: Define the variable of interest and the population(s) of interest. Step 2: Determine if the sample represents the population of interest Step 3: Determine if the problem is an estimation problem or hypothesis test problem Step 4: If the research question requires you to perform a hypothesis test, state the null and alternative hypotheses in notation and words. Step 5: Perform an exploratory analysis of the sample data. Include graphs and summary information. When working with categorical data, consider creating a bar chart to display the data. The summary statistics for a problem like this would be the total sample size and the number of individuals in each category. Remember that categorical data does not have a mean or a standard deviation. Step 6: Conduct a hypothesis test and determine a p-value

Formal Steps for Inference:

Step 1: Define the variable of interest and the population(s) of interest. Determine if the variable of interest is quantitative or categorical and how many populations there are. Step 2: If a sample was taken or an observational study performed, assess whether the data collected are representative of the data in the population to which the researcher wants to make a conclusion. If an experiment was performed, were the experimental units randomly assigned to the different "treatment" groups? Step 3: Determine if the problem is an estimation problem or hypothesis test problem. If the problem is an estimation problem: Step 4: Perform an exploratory analysis of the sample data to get an idea of a good estimate of the population parameter. Step 5: Obtain a confidence interval using either the formula method or the percentile method using bootstrapping using either the formula method or the percentile method Step 6: Interpret the confidence interval in the context of the problem. Comment if this conclusion is valid to the population of interest. If the problem is a hypothesis test problem: Step 4: State the null and alternative hypotheses in notation and words. Step 5: Perform an exploratory analysis of the sample data to get an idea of whether or not the sample data support the claim made in the alternative hypothesis. Step 6: Determine the p-value. Step 7: Use the p-value and the exploratory analysis to state a conclusion in the context of the problem. Comment if this conclusion is valid to the population of interest.

The process to determine a p-value using the bootstrap methods is as follows:

Step 1: Determine what sample statistics are considered as or more unusual than the observed sample statistic if the null hypothesis is true. Step 2: Create a bootstrap distribution of sample statistics using data such that the null hypothesis is true. Step 3: Obtain the p-value by finding the proportion of bootstrap sample statistics from the bootstrap distribution created in Step 2 that are considered as or more unusual.

To perform a hypothesis test using two-sample t-methods, we proceed with the following seven steps:

Step 1: Identify the variable of interest and the populations Step 2: Assess if the samples are representative of their respective populations Step 3: Determine if this is a hypothesis test or estimation problem Step 4: State the null and alternative hypotheses Step 5: Explore the sample data Step 6: Determine the p-value Step 7: Answer the question of interest

What is the Correct Parameter to Use? (Four Steps)

Step 1: Identify the variable of interest and the type of variable • If the variable of interest is quantitative, the parameter used will be μ. If the variable of interest is categorical with two categories, the parameter will be p. If the problem requires you to compare a quantitative variable between two groups, the parameter will be μgroup1 − μgroup2 Step 2: Identify the population or populations of interest • If there is one population, the null hypothesis is of the form: parameter = hypothesized value. If there are two populations, the null hypothesis will be of the form parameter for group 1 = parameter for group 2. In this class, we will only discuss a situation when we compare two populations means. Step 3: In the context of the problem, define the parameter that will be used in the hypotheses Step 4: Write the hypotheses • Be able to write the hypotheses in notation AND in words in the context of the problem

In a statistical hypothesis test, the evidence will be the sample data. Therefore, in a statistical hypothesis test, when are the hypotheses determined? a) before data are collected b) after data are collected c) as data are being collected d) the hypotheses can be determined at any time before, during, or after data are collected

a) before data are collected

It's been suggested that college students spend 2 to 3 hours outside of class studying for every one credit hour. So, for a 4 credit course, students spend 4 hours in class each week and then another 8 to 12 hours per week outside of class on the course. Suppose a faculty member wanted to know if students actually spend the appropriate amount of time outside of class studying, as she feels students are not spending enough time outside of class on her course. For her 4-credit class, she expected students to spend 8 hours outside of class each week on the course (reading, studying, assignments, etc.), on average. She randomly sampled 35 students from her very large lecture and found the sample mean amount of time spent on the course outside of class each week was 7.3 hours. Determine the correct parameter to use and write the hypotheses:

Step 1: The variable of interest is what is recorded on each case: time (in hours) spent on this particular course outside of the classroom each week. It is quantitative. Step 2: The population is all students in this faculty member's 4-credit class. There is one population. Step 3: The parameter is µ = the mean amount of time spent on this faculty member's 4-credit course outside the classroom each week (in hours) Step 4: There are two competing hypotheses in this problem: 1. Students spend the recommended 8 hours per week, on average, doing work outside of this 4-credit class. 2. Students spend less than the recommended 8 hours per week, on average, doing schoolwork outside of this 4-credit class. This is what the faculty member believes may be happening.

Below you will find descriptions of part of the process of hypothesis testing using the t-methods. Please identify each part as belonging to: Step 1: Define the variable of interest and the population of interest. Step 2: Discuss if the sample represents the population of interest. Step 3: Decide if you need to conduct a hypothesis or calculate a confidence interval. Step 4: If a hypothesis is to be performed, define the null and alternative hypothesis. Step 5: Explore the data to verify that the conditions of the test have been satisfied Step 6: Perform the formal analysis Step 7: State the conclusions of the study based on the formal a) Since the days in which they recorded data was randomly selected, they believe their sample represents the population of daily change in stock price for this given stock. b) Researchers obtain a histogram and summary statistics of the data collected. They notice that the distribution of changes in daily stock prices is roughly normally distributed. c) Researchers record the changes in stock prices for a single stock for 10 randomly selected days within a period. They hope to perform a hypothesis to determine if the average daily change for this stock is more than $5. d) Researchers conclude there is strong evidence that the mean daily change in stock price for this particular stock is greater than $5. e) Researchers plan on conducting a hypothesis test to answer their question of interest f) They establish a null hypothesis that the mean daily change in stock price is $5. They hope to show that the data suggest the mean daily price in stock is more than $5. g) A hypothesis test is conducted. Researchers obtain a t-statistic of 2.3 on 34 degrees of freedom and a p-value of 0.014.

Step 1: c) Researchers record the changes in stock prices for a single stock for 10 randomly selected days within a period. They hope to perform a hypothesis to determine if the average daily change for this stock is more than $5. Step 2: a) Since the days in which they recorded data was randomly selected, they believe their sample represents the population of daily change in stock price for this given stock. Step 3: e) Researchers plan on conducting a hypothesis test to answer their question of interest Step 4: f) They establish a null hypothesis that the mean daily change in stock price is $5. They hope to show that the data suggest the mean daily price in stock is more than $5. Step 5: b) Researchers obtain a histogram and summary statistics of the data collected. They notice that the distribution of changes in daily stock prices is roughly normally distributed. Step 6: g) A hypothesis test is conducted. Researchers obtain a t-statistic of 2.3 on 34 degrees of freedom and a p-value of 0.014. Step 7: d) Researchers conclude there is strong evidence that the mean daily change in stock price for this particular stock is greater than $5.

How do you know you are dealing with a proportion problem?

The biggest give away is the problem does not mention any mean or standard deviation. Instead, the problem talks about a proportion.

Interval Estimate

The list (or "range") of possible values for the population parameter Ex. An interval estimate for average heights of all fifth grade girls might be from 52 to 56 inches

Law of Large Numbers

The long-run proportion of trials repeated under identical conditions will get closer to the true proportion as the number of trials increases

Every bootstrap sample generated could be different from one to the next since we are resampling with replacement. Therefore, the value of each bootstrap sample mean could be different as well. Recall that our original sample was 6, 11, 17, 22 and 29 months. The minimum possible value for a bootstrap sample mean is _____ months and the maximum possible value for a bootstrap sample mean is _____ months

The minimum possible value for a bootstrap sample mean is 6 months and the maximum possible value for a bootstrap sample mean is 29 months

Sampling Without Replacement

The object selected is NOT replaced into the pool before the next object is selected, which means it cannot be selected again

Sampling with Replacement

The object selected is replaced into the "pool" before the next object is selected so that it's possible that object could be selected again

Equally Likely Outcomes

The outcomes in a sample space all have the same probability of occurring

The Oregon Department of Transportation is interested in the average number of miles driven by working adults aged 35 - 46 in a one week period. ODOT believes that working adults in this age range drive more than 100 miles in a one week period. ODOT sends a survey to 1000 such adults and surveys are returned by 274 adults. The survey asks respondents for an estimate of how many hours they believe they drive each week. ODOT determines that the average number of miles driven each week by the respondents of the survey is 74.9 miles. Use the correct p-value (.0154) from the last question to make a decision and write a conclusion. Fill in the blanks to write a correct conclusion: There is (strong, some, weak, or not sufficient) evidence to indicate the (number of miles driven mean number of miles driven or mean number of miles driven) by all working adults ages 35 - 46 in a one week period is (equal to, not equal to, less than, or more than, or less than) (74.9 or 100).

There is some evidence to indicate the mean number of miles driven by all working adults ages 35 - 46 in a one week period is less than, or more than 100.

Constructing a Confidence Interval

To determine the lower and upper bounds of the confidence interval

Resampling

To take a sample with replacement of the same sample size as the original sample from the original sample • Such a sample is called a bootstrap sample • As long as the data from the sample are representative of the data from the population of interest, we can resample from our sample to estimate the sampling distribution and, therefore, estimate what the exact standard deviation of the sampling distribution is.

Inference

To use data from a sample to make a generalization or conclusion about the population from which it came

True of False? Data can be statistically significant, but not be practically significant.

True

True of False? Researchers can make any difference between the observed statistic and the hypothesized value statistically significant by taking a large enough sample size, no matter how small that difference may be.

True

True of False? When reading an article, we should always consider the sample size in addition to a p-value when assessing the authors' conclusions.

True

True or False? If a sample size is very small (say less than 10) and the bootstrap distribution of sample means is skewed, it may not be appropriate to construct a confidence interval for the mean or the median.

True

True or False? If all unique random samples of the same size from some population are listed, the sample means from all the random samples form an exact sampling distribution.

True

True or False? Most random samples will yield sample means of different values. In other words, there is variation among the sample means.

True

True or False? Statistical significance means there was evidence to reject the null hypothesis in a statistical hypothesis test.

True

True or False? Suppose two random samples are generated from the same population, one with n = 10 and one with n = 100. Bootstrap distributions are generated for each sample size. The standard deviation of the bootstrap distribution generated from the sample with n = 100 will be less than the standard deviation of the bootstrap distribution generated from the sample with n = 10.

True

True or False? The sample means are centered at the population mean.

True

True or False? The sample means can be plotted in a histogram since sample means are quantitative.

True

True or False? The shape of a bootstrap distribution resembles the shape of the distribution of sample means.

True

True or False? When a bootstrap distribution of sample means is skewed, the formula method of determining the bounds should not be used.

True

True or False? A p-value is a probability.

True

True or False? The pth percentile is the value such that p percent of the remaining observations are less than it.

True

True or False? The sampling distribution of the sample means is always centered at the population mean.

True

True or False? The spread in the sample means decreases when larger sample sizes are taken.

True

True or False? A decision that can be made in a hypothesis test is that the null hypothesis is false and can be rejected

True - When conducting a hypothesis test we collect evidence to see if we are able to support the alternative. If we are able to conclusively support the alternative hypothesis, then in that same conclusion we are able to reject the null hypothesis

True - When tossing a coin twice, we cannot have both tosses be heads AND both tosses be tails

True or False? 1. The spread of the distribution of sample means will always be less than the spread in the population regardless of the sample size.

True - the range of sample means will have to be less than the range of the population data; true for any measure of the spread used

True or False? If the population data are extremely skewed, it may take a sample size in the hundreds before the distribution of sample means is approximately normal.

True - the shape of the population data to have a rough idea of what sample size is needed for the sample means to be approximately normal

True or False? Sampling distributions can be created for any statistic, not just sample means.

True - there is a sampling distribution for any statistic

Data were collected in North Carolina over the period of a year on 150 randomly selected mothers and their newborn babies. Although many variables were measured, researchers were interested in whether or not the mother was a smoker and the weight of their newborn baby. Researchers would like to know if there was evidence that newborns from mothers who smoke have a different average birth weight than newborns from mothers who don't smoke. The following summary statistics were found on each group of newborns. Consider the samples of newborns as representative of all newborns born to smoking and nonsmoking mothers. Smoker mean 6.78 standard deviation 1.43 sample size 50 Nonsmoker mean 7.18 standard deviation 1.6 sample size 100 Do the data provide evidence of a difference in birth weight for babies born to mothers who smoke and mothers who do not? Use a significance level = 0.05. Now it is time to state the conclusion of the test. Using the correct information from the previous questions (Ha: μn− μs ≠ 0, t-statistic = 1.551, df = 49, p-value = 0.1273, α = 0.05) complete the sentence. With a p-value of 0.1273, we (accept, reject, fail to reject) the null hypothesis at the (0, 49, 0.1273, 0.05, 1.551) significance level. The data provide (not enough, strong moderate) evidence to suggest that the mean birth weight of babies born to mothers who do not smoke (is equal to, is less than, is different than, is greater than) the mean birth weight of babies born to mothers who do smoke.

With a p-value of 0.1273, we (fail to reject) the null hypothesis at the (0.05) significance level. The data provide (not enough) evidence to suggest that the mean birth weight of babies born to mothers who do not smoke (is different than) the mean birth weight of babies born to mothers who do smoke.

A pharmaceutical company has developed a new drug they believe will help relieve symptoms associated with Crohn's Disease. A clinical trial involving 50 patients under the age of 30 with Crohn's Disease was conducted to compare the effectiveness of the new drug at relieving such symptoms compared to the standard drug. Is this scenario an example of inference or not?

Yes, It is an example of inference as the pharmaceutical company hopes to market the new drug to all patients with Crohn's Disease. Therefore, the patients in the clinical trial are a sample of all patients with Crohn's Disease.

Suppose that a confidence interval is desired for a certain observational study. The point estimate is 4.5 feet and the margin of error is 0.85. Which of the following is the correct confidence interval? a) (3.65 feet, 5.35 feet) b) 2.80, 6.20 c) 3.65, 5.35 d) (2.80 feet, 6.20 feet)

a) (3.65 feet, 5.35 feet)

Suppose that the following confidence interval is obtained from a study: (0.45 cm, 1.34 cm). Which of the following values would be rejected as a plausible value for the population parameter of interest if it was tested in the null hypothesis? (Check all that apply) a) 0.1 cm b) 2 cm c) 0.46 cm d) -1.56 cm e) 1 cm

a) 0.1 cm b) 2 cm d) -1.56 cm

a) 1 • Only one sample was taken - a sample of 40 consumers at this mall.

A deck of playing cards has 52 cards, divided into four suits (diamonds, spades, clubs, and hearts). There are an equal number of cards in each suit in this deck. If three cards are drawn, what is the probability that the first two cards drawn are clubs and the last is red? There are three events in this problem: Let event A = first card drawn is a club Let event B = second card drawn is a club C = third card drawn is red. We want to find P(A and B and C) (i.e. the probability that the first card drawn is a club AND the second card drawn is a club AND the third card drawn is a red card). Suppose the first card drawn is a club and is not replaced into the deck before the second card is drawn. Also, suppose event A occurred (first card drawn is a club). What is the probability that the second card drawn is a club? a) 12/51 b) 12/52 c) 13/51 b) 13/52

a) 12/51

What is the percentile of the lower bound of a 95% confidence interval? a) 2.5th b) 5th c) 10th d) 95th e) 97.5th

a) 2.5th

A deck of playing cards has 52 cards, divided into four suits (diamonds, spades, clubs, and hearts). There are an equal number of cards in each suit in this deck. If three cards are drawn, what is the probability that the first two cards drawn are clubs and the last is red? There are three events in this problem: Let event A = first card drawn is a club Let event B = second card drawn is a club C = third card drawn is red. We want to find P(A and B and C) (i.e. the probability that the first card drawn is a club AND the second card drawn is a club AND the third card drawn is a red card). Suppose that the first card drawn is a club and is not replaced into the deck before the second card is drawn. Suppose also that the second card drawn is a club and is not replaced into the deck before the third card is drawn. What is the probability that the third card drawn is red? a) 26/50 b) 24/52 c) 24/50 d) 25/51 e) 1/2

a) 26/50 - The first two cards drawn are clubs, which are black cards. They are not replaced, so there are only 50 cards left in the deck. The deck started with 26 red cards and still has 26 red cards.

Five samples were taken for a free dental exam, responses in months since last exam. The original sample was 6 11 17 22 29 Identify which of the following are legitimate bootstrap samples a) 6, 17, 11, 22, 29 b) 6, 6, 6, 6, 6 c) 17, 22, 6, 17, 22 d) 3, 6, 11, 11, 11 e) 29, 11, 17, 6, 6, 22 f) 11, 17, 6 g) 11, 11, 17, 17, 11

a) 6, 17, 11, 22, 29 - Yes b) 6, 6, 6, 6, 6 - Yes c) 17, 22, 6, 17, 22 - Yes d) 3, 6, 11, 11, 11 - No e) 29, 11, 17, 6, 6, 22 - No f) 11, 17, 6 - No g) 11, 11, 17, 17, 11 - Yes

The Oregon Department of Transportation is interested in the average number of miles driven by working adults aged 35 - 46 in a one week period. ODOT believes that working adults in this age range drive less than 100 miles in a one week period. ODOT sends a survey to 1000 such adults and surveys are returned by 274 adults. The survey asks respondents for an estimate of how many hours they believe they drive each week. ODOT determines that the average number of miles driven each week by the respondents of the survey is 74.9 miles. Which of the following quantities should we subtract from each sample value in order to center the bootstrap distribution of the sample means at the hypothesized value? a) 74.9 - 100 = -25.1 b) 1000 - 274 = 726 c) 35 - 46 = -11 d) 100 - 74.9 = 25.1

a) 74.9 - 100 = -25.1 • In order to have a bootstrap distribution centered at the hypothesized value we should subtract off the (sample mean - the hypothesized value). The sample mean is 74.9 which was calculated using the sample data obtained from the survey and the hypothesized value is 100 because this is what ODOT believes to be true.

Suppose there are 5 people in a room and we are interested in the average shoe size of people in the room. We randomly sample 2 people from the room and take the average of their shoe sizes. We repeat this process again and again until every possible unique combination of two people has been used and we take the values and construct a sampling distribution. Suppose the shoe sizes of the 5 people are 10, 8, 11, 12, and 9. What will be the values of the sample statistics that make up the sampling distribution? a) 9, 10.5, 11, 9.5, 9.5, 10, 8.5, 11.5, 10, 10.5 b) 18, 21, 22, 19, 19, 20, 17, 23, 20, 21 c) 13, 2, 9, 11, 12, 19, 21, 19, 23, 18

a) 9, 10.5, 11, 9.5, 9.5, 10, 8.5, 11.5, 10, 10.5 - Construct a table similar to that used in the last question but replace the person's number with their shoe size. The cells are the means now and we still only calculate the means for the white cells. The table is filled out below with some cells showing full calculations.

Which confidence interval is narrowest? a) 90% confidence interval b) 95% confidence interval c) 99% confidence interval

a) 90% confidence interval

Which of the following best matches the definition of a sampling distribution? a) A sampling distribution is a list of the values for sample statistics from random samples of the same size from the same population. b) A sampling distribution is a list of values from our sample. c) A sampling distribution is a list of the population parameters for every possible population

a) A sampling distribution is a list of the values for sample statistics from random samples of the same size from the same population.

Suppose that in a jury trial, the jury gives the judge the verdict "not guilty." In the context of the problem, which of the following is the correct interpretation of the error made? a) Decide the defendant is not guilty when in fact the defendant is guilty. b) Decide the defendant is not guilty when the defendant is innocent. c) Decide the defendant is guilty when in fact the defendant is innocent. d) Decide the defendant is guilty when the defendant is indeed guilty.

a) Decide the defendant is not guilty when in fact the defendant is guilty. • Making a Type II Error is letting a guilty person go free. (Decide the defendant is not guilty when in fact the defendant is guilty.)

Allison wonders what the average starting salary for business majors in their first job out of college. She is obtains a random sample of recent business graduates from her school. This scenario is an example of which type of problem? a) Estimation Problem b) Hypothesis Test Problem

a) Estimation Problem

H0: p = 0.5 HA: p ≠ 0.5 90% confidence interval for p: (0.48, 0.74) Using the confidence interval, what decision should be made? a) Fail to reject the null hypothesis b) Reject the null hypothesis

a) Fail to reject the null hypothesis • The hypothesized value (0.5) is between the bounds, which indicates it is one of the possible values for the population proportion. Therefore, we do not want to reject the null hypothesis.

H0: μ1 = μ2 HA: μ1 ≠ μ2 99% confidence interval for μ1 - μ2: (-0.25, 1.43) Using the confidence interval, what decision should be made? a) Fail to reject the null hypothesis b) Reject the null hypothesis

a) Fail to reject the null hypothesis • The null hypothesis can also be written H0: μ1 - μ2 = 0. The hypothesized value (0) is between the bounds, which indicates it is one of the possible values for the difference in population means. Therefore, we do not want to reject the null hypothesis.

Marketing companies have collected data implying that teenage girls use more ring tones on their cellular phones than teenage boys do. In one particular study of 20 randomly chosen teenage girls and 20 randomly chosen teenage boys with cellular phones, the mean number of ring tones for the girls was 3.2 with a standard deviation of 1.5. The mean for the boys was 1.7 with a standard deviation of 0.8. Conduct a hypothesis test to determine if the means number of ringtones on girls' phones is greater than for boys. What is null and alternative hypothesis? a) H0: µGirls = µBoys HA: µGirls > µBoys b) H0: µGirls = µBoys HA: µGirls < µBoys c) H0: µGirls < µBoys HA: µGirls > µBoys d) H0: µGirls = µBoys HA: µGirls ≠ µBoys

a) H0: µGirls = µBoys HA: µGirls > µBoys

Select all options below that show incorrect notation for an alternative hypothesis? (Select all that apply) a) HA: μ = 0 b) HA: μ < 0 c) HA: μ > 0 a) HA: μ ≠ 0

a) HA: μ = 0 • The options containing a sign that reflects a range of values for the population parameter are all possibilities for the alternative hypothesis.

Consider the process of flipping a fair coin three times. What is the sample space of this random experiment? For the answer choices assume that H = heads and T = tails. a) HHH, HTH, THH, TTH, HHT, HTT, THT, TTT b) HHH, HTH, TTT, THT c) HHH, TTT, THH, TTH, HHT, HHH, TTT, THH

a) HHH, HTH, THH, TTH, HHT, HTT, THT, TTT • There are 2^3 = 8 possible outcomes since we have two outcomes on each flip and three flips. None of the outcomes should be repeated twice in our sample space.

Researchers at the World Health Organization are interested in the mean height of elephants in sub-Saharan Africa. They randomly sample 1000 elephants and calculate their height in feet. With this data they construct the following 95% confidence interval: (9.52 feet, 12.60 feet). Which of the following is the correct interpretation of the confidence interval? a) We are 95% confident that the mean height of elephants in sub-Saharan Africa is between 9.52 feet and 12.60 feet b) There is a 95% chance that an elephant in sub-Saharan Africa will be between 9.52 feet and 12.60 feet c) We are 95% confident that the height of elephants in sub-Saharan Africa is between 9.52 feet and 12.60 feet

a) We are 95% confident that the mean height of elephants in sub-Saharan Africa is between 9.52 feet and 12.60 feet

Which argument concerning sampling with replacement versus sampling without replacement is correct in the context of bootstrapping? a) If we sample from the same original sample without replacement and our sample is the same size as the original sample we will just obtain repeated samples that are exactly the same. Therefore we should sample with replacement. b) We should sample without replacement but only if our original sample is over size 30. Then it is representative of the population and we can duplicate it to obtain an estimate of the sampling distribution. c) The original sample is representative of the population so we need to duplicate that same sample so that all of our samples are representative of the population. Therefore we should sample without replacement.

a) If we sample from the same original sample without replacement and our sample is the same size as the original sample we will just obtain repeated samples that are exactly the same. Therefore we should sample with replacement.

Suppose an internet marketing company wants to determine the number of times an ad is presented to a user on Facebook before they click on that add. One hundred Facebook users were randomly selected and the same advertisement was added to their feeds. The number of times the user scrolled past the advertisement before clicking on it was recorded. Can researchers use one-sample z-methods for proportions to analyze these data? a) No - the data is quantitative and not categorical. b) No - we cannot assume the sample is representative of the population. c) Yes - all of the conditions have been met so researchers can use one-sample z-methods. d) No - the sample size is not large enough.

a) No - the data is quantitative and not categorical. • The number of times a user scrolls past an ad before clicking is a quantitative variable and we should not use z-methods for a population proportion.

Allison wonders what the average starting salary for business majors in their first job out of college. She is obtains a random sample of recent business graduates from her school. From the sample, Allison will estimate the average starting salary for all business majors in their first job out of college. Therefore, Allison is estimating a? a) Parameter b) Statistic

a) Parameter

Which of the following elements of a study impact the width of a confidence interval? (Select all that apply) a) Sample size b) Units of the variable of interest c) Confidence level d) Parameter of interest

a) Sample size c) Confidence level

Which of the following is true regarding the bounds of the 95% confidence interval for μ if the percentile method is used? a) The bounds would be nearly the same as the bounds from the formula method since the bootstrap distribution is approximately normal. b) The bounds would be nearly identical as the bounds from the formula method since the bootstrap distribution is centered at the sample mean. c) The bounds may be very different than the bounds from the formula method, or they might be the same - it just depends on the bootstrap samples taken. d) The bounds would be very different than the bounds from the formula method since the bootstrap distribution is centered at the hypothesized value.

a) The bounds would be nearly the same as the bounds from the formula method since the bootstrap distribution is approximately normal.

If the null hypothesis is true, where should the bootstrap distribution of sample means be centered? a) The hypothesized value of the population mean b) The observed sample mean

a) The hypothesized value of the population mean

Based on this probability of 0.0039, what decision would you make about the lady's ability to identify the preparation method of tea? To clarify, the lady will have a 0.39% chance of guessing the correct preparation method in 8 of the 8 cups IF she was simply guessing. Since this probability is so small, do you think that she may be able to detect the preparation method? a) The p-value is quite low, which indicates that it is very unlikely for her to correctly identify the preparation method on 8 cups if she was just guessing on each. Therefore, I would say the alternative hypothesis is true - she's not just guessing but can really distinguish between the tastes of the two preparation methods. b) The p-value is quite high, which indicates that it is very likely for her to correctly identify the preparation method on 8 cups if she was just guessing on each. Therefore, I would say the null hypothesis is true - she is just guessing and can't really distinguish between the tastes of the two preparation methods. c) The p-value is quite low, which indicates that it is very likely for her to correctly identify the preparation method on 9 (or more) cups if she was just guessing on each. Therefore, I would say the null hypothesis is true - she is just guessing and can't really distinguish between the tastes of the two preparation methods. d) The p-value is quite high, which indicates that it is very unlikely for her to correctly identify the preparation method on 8 cups if she was just guessing on each. Therefore, I would say the alternative hypothesis is true - she's not just guessing but can really distinguish between the tastes of the two preparation methods.

a) The p-value is quite low, which indicates that it is very unlikely for her to correctly identify the preparation method on 8 cups if she was just guessing on each. Therefore, I would say the alternative hypothesis is true - she's not just guessing but can really distinguish between the tastes of the two preparation methods.

How does the spread of the bootstrap distribution compare to the spread of the original sample data? To answer this question you may consider comparing standard deviations. a) The spread of the bootstrap distribution is less than the spread of the sample data. b) The spread of the bootstrap distribution is more than the spread of the sample data. c) The spread of the bootstrap distribution is the same as the spread of the sample data. d) This cannot be determined without a graph of the sample data.

a) The spread of the bootstrap distribution is less than the spread of the sample data.

A basketball player shoots two free throws in a row. Let A = she makes the first free throw. Let B = she misses the second free throw. What would be an argument that the two events are not independent? a) There may be a psychological effect - if she makes the first attempt, she may be more confident in her ability to make the second attempt. b) Both events can occur in one repetition of this "experiment". c) Exactly one of the events must occur in one repetition of this "experiment". d) The probability of making a free throw will remain the same regardless of whether or not she made the previous attempt.

a) There may be a psychological effect - if she makes the first attempt, she may be more confident in her ability to make the second attempt.

A political science professor wondered if there was a difference in the proportion favoring a recent U.S. Supreme Court Justice nominee between voters who identify themselves as Republican and voters who identify themselves as Democrats. She randomly sampled approximately 3000 Democrats and 3000 Republicans from voter registration cards for each party. Which is the following is correct? a) This is an inference problem that involves performing a hypothesis test. b) This is an inference problem that involves estimation only. c) This is not an inference problem.

a) This is an inference problem that involves performing a hypothesis test.

Suppose α = 0.10, β = 0.13, and p-value = 0.03 What error could have been made? a) Type I Error b) Type II Error

a) Type I Error • Because a decision was made to reject the null hypothesis, a Type I Error could have been made

Which of the following is the correct formula for calculating the width of a confidence interval? a) Width = upper bound - lower bound b) Width = lower bound - upper bound

a) Width = upper bound - lower bound

28. The National Center for Education Statistics monitors many aspects of elementary and secondary education nationwide. Their 1996 numbers are often used as a basis to assess change. In 1996, 31% of students reported that their mothers had graduated from college. Suppose in 2015, responses from 8368 students found 2929 indicated their mothers had graduated from college. Is this evidence of a change in education level among mothers? a. What are the null hypothesis and the alternative hypothesis in statistical notation? • Null: • Alternative: b. Is the sample size large enough to assume that the sampling distribution of 𝑝𝑝̂ is approximately normal. Explain c. Calculate the test statistic and degrees of freedom (if necessary) for this hypothesis test. d. A p-value was found to be < 0.0001. Write your conclusions in the context of this study using a significance level of 0.05.

a. • Null: H0: p = .31 • Alternative: HA: p > .31 b. Yes. We have at least 10 successes and 10 failures. The sample is very large. c. 𝑝̂ = 2929/8368 = 0.35 z = 5.46 d. There is strong evidence to suggest that the true proportion of mothers that had graduated college by 2015 is greater than 0.31.

27. Isabel Myers was a pioneer in the study of personality types. She identified four basic personality preferences that are described in her book A guide to the Development and Use of Myers-Briggs Type Indicator. Marriage counselors know that couples who have none of the four personality traits in common have a stormy marriage. a. Myers took a random sample of 375 married couples and found that 298 had two or more personality traits in common. Using this information Construct a 95% confidence interval for the proportion of all married couples that have at least two personality traits in common (assume these data represent all married couples). Round your final answer to 2 decimal places and report your interval in correct notation. Don't forget units. b. Interpret the confidence interval in the context of the problem.

a. (0.7537, 0.8357) b. With 95% confidence, the proportion of married couples with at least two personality traits in common is between 0.7537 and 0.8357.

24. A survey was conducted by Greenfield Online, 25-34 year olds spend the most each week on fast food. Based on a sample of 55 participants, the average weekly amount of $44 with a standard deviation of $14.50. This was reported in a May 2009 USA Today Snapshot. a. Construct a 90% confidence interval for the mean weekly amount that 25-34-year-olds spend each week on fast food. a. Interpret the confidence interval found in part a in the context of the problem. c. Suppose you and your friend Larry were discussing the results of this study and Larry claims that he believes the average amount that 25-34 year-olds spend on fast food is $35. Using the confidence intervals calculated in part a, how would you respond to Larry's statement? That is, do you believe Larry is correct or incorrect in stating that the average amount spent is $35?

a. (44) ± (1.676)14.5√55 44 ± (1.676)(1.95518) 44 ± 3.27688 ($40.72, $47.28) b. With 90% confidence, the true mean amount spent each week on fast food by 25-34-year-olds is between $40.72 and $47.28. c. Since a 90% confidence interval suggests the average amount that 25-34 year-olds spend on fast food each week is between $41.76 and $46.22, Larry's claim of $35 seems to be incorrect.

a. Experiment, but not double-blinded.

9. Which of the following statements is true regarding the relationship between a 90% and a 95% confidence interval based on the same sample? a. The 90% confidence interval will have a smaller range compared to the 95% confidence interval. b. The 90% confidence interval will have a larger range compared to the 95% confidence interval. c. The confidence level doesn't matter. The two intervals will be exactly the same since they are based on the same sample. d. It is impossible to know without having the values of the sample mean and standard deviation.

a. The 90% confidence interval will have a smaller range compared to the 95% confidence interval.

25. The alcohol content of wine depends on the grape variety, the way in which the wine is produced from grades, the weather, and other influences. However, the alcohol content of wine varieties must remain somewhat consistent. The distribution below represents the percent of alcohol in wine produced from a random sample of 48 wine makers in the same region of Italy. The sample yields a mean and standard deviation of 𝑥̅ = 13.39 percent and s = 0.91 percent, respectively. The alcohol content of this varietal has a target of 13%, but it is common that wines contain more alcohol than their target. Is there evidence the alcohol content of the wine from this region is greater than 13%? Use a significance level of 0.01. a. Fully describe the distribution using context and terms for the shape, center and spread of the distribution. Is there visual evidence the true mean may be different than 13%? b. State the null and alternative hypothesis for the appropriate hypothesis test. c. Calculate the test statistic and p-value for the hypothesis test and degrees of freedom. d. Calculate the 99% confidence interval for 𝜇. e. Using your answers from above, summarize your conclusions in the context of the problem. Include a statement in terms of the amount of evidence the data provides in support of the alternative, as well as state and interpret the confidence interval and point estimate. Use a significance level of 0.05. f. How would changing the significance level from 0.01 to 0.05 affect the following values in this problem? Fill in the blank for each statement using: "Decrease" "Not change" "Increase" • The test statistic will _____. • The critical value will _____. • The p-value will _____. • The margin of error will _____.

a. The percent of alcohol content is centered a little above 13 percent, with slight skew in the positive direction towards higher alcohol contents. The percent of alcohol ranges approximately from 11.5 to 16%. b. Ho: 𝝁 = 13 percent , HA: 𝝁 > 13 percent c. t = 2.97 = (13.39−13) / (0.91√48) T table: 0.001< p-value <0.005 Software P-value = 0.0023 df = 47 d. 13.39 ± 2.704 * 0.91/sqrt (48) 13.39 ± 2.704* 0.1313 13.39 ± - 0.3550 99% CI (13.035, 13.745) e. There is convincing evidence the average alcohol content in the wine is greater than 13%. The sample estimates the average content of alcohol is 13.39% with a 99% confidence interval from 13.04% to 13.75%. The null hypothesis is rejected. A one-sided t-test resulted in a significant p-value of 0.002 from a test statistic of 2.97 with 47 df.(4) f. How would changing the significance level from 0.01 to 0.05 affect the following values in this problem? Fill in the blank for each statement using: "Decrease" "Not change" "Increase" • The test statistic will Not change. • The critical value will Decrease. • The p-value will Not Change. • The margin of error will Decrease.

7. A study was performed to determine if there is a difference in the average amount of coffee consumed between male and female students between the ages of 18 to 25. This study found a two-sided p-value of 0.34. Therefore, researchers concluded there is strong evidence to suggest that there is no difference between the average amount of coffee consumed by males and females. Which of the following is true regarding the description of this study? a. The researchers wrongfully made a conclusion that accepted the null hypothesis. b. For a two sample test, researchers only needed to find a one-sided p-value. c. A one-sample t-test must have been used to find the p-value. d. The variable of interest in this study is whether or not a person drinks coffee.

a. The researchers wrongfully made a conclusion that accepted the null hypothesis.

4. Natalie has just finished her term data project and was able to find evidence to support her alternative hypothesis and therefore rejected the null hypothesis using a significance level of 0.05. She conducted the study in such a way so that the power of her test was 0.85. Based on her conclusion, what kind of error could she have made and what is the probability of this error? a. Type I and P(Type I) = 0.05 b. Type I and P(Type I) = 0.15 c. Type I and P(Type I) = 0.85 d. Type II and P(Type II) = 0.05 e. Type II and P(Type II) = 0.85

a. Type I and P(Type I) = 0.05

23. A report indicated that automobiles manufactured in North America are less fuel-efficient (measured in miles per gallon) than automobiles manufactured in Asia. To test this, researchers gathered a random sample of 25 cars manufactured in the United States and the fuel efficiency was determined for each vehicle. Please find the summary statistics and graphical displays of the data below. Mean fuel efficiency (miles per gallon) 28.9 Sample standard deviation 3.25 Sample size 25 a. What is the variable of interest in this study? b. Suppose research indicated that the mean fuel efficiency for cars manufactured in Asia is 31 miles per gallon (mpg). We are interested to see if the average fuel efficient for cars made in America is less than 31 mpg. What are the null hypothesis and the alternative hypothesis in statistical notation? • Null: • Alternative: c. In this study, will the distribution of sample means be approximately normal? Why or why not? Please reference what information or graphical displays you used to support your answer. d. Calculate the test statistic and degrees of freedom for this hypothesis test. Show all work. e. Write your conclusions in the context of this study. With your conclusion, please provide the p-value for the study.

a. fuel efficiency or miles per gallon b. • Null: H0: µ = 31mpg • Alternative: HA: µ < 31 mpg c. The histogram shows approximately normal sample data. Since the sample data is approximately normal, the assumption is that the population data is normal, therefore we should feel comfortable stating the distribution of differenced in sample means will be the same. Since the sample size is 25, it is on the border of being "large enough." d. Degrees of freedom = 24 = 25-1 t = -3.2 = (28.9-31) / (3.25/√25) e. There is strong evidence to indicate the fuel efficiency for cars manufactured in the U.S. is less than the fuel efficiency of cars manufactured in Asia. (p-value < 0.005)

To center the bootstrap distribution of sample means at the hypothesized value, we will _____ the sample data by adding the difference between the hypothesized value and the sample mean from each value in the sample.

adjust

The sample data are used to assess whether we feel there is evidence to support the claim in the ______ hypothesis and, therefore, reject the claim in the ______ hypothesis

alternate, null

In order to reject the null hypothesis, you need to be convinced beyond a reasonable doubt that the ______ hypothesis is true.

alternative

The ______ hypothesis will either be <, >, or ≠. Which one is used depends solely on the question of interest.

alternative

When answering the question of interest, we must state how much evidence we have in favor of the _______ hypothesis.

alternative

The ______ hypothesis is often associated with the claim a researcher is making or what a researcher is wondering. This is the claim the researcher often hopes the evidence will support.

alternative Ex. If a study was being conducted to test the effectiveness of a new drug compared to the old drug, the researchers are hoping that the new drug will be more effective than the old drug. Therefore, the alternative hypothesis is that the new drug is more effective than the old drug.

If the sample data is more consistent with the claim made in the ______ hypothesis then we may have evidence to support that ______ hypothesis and say that it is true

alternative, alternative

Select the option that best completes the following sentence: A p-value is the probability of observing a test statistic that is (greater, less, as or more unusual) than the one observed if the null hypothesis is true.

as or more unusual

Suppose α = 0.10, β = 0.13, and p-value = 0.03 What is the probability of making that error? a) .03 b) .10 c) .13 d) .87 e) .90 f) .97

b) .10 • P(Type I Error) = α = .10

Suppose α = 0.05, power = 0.85, and p-value = 0.3 What is the probability of making that error? a) .05 b) .15 c) .30 d) .85 e) .95

b) .15 • P(Type II Error) = 1 - power = 1 - .85 = .15

Here is the output from one simulation of 10,000 bootstrap samples: 2.5% 98.13667 97.5% 98.53000 Therefore, the 95% confidence interval for μ, the mean body temperature for all students, is (98.14ºF , 98.53 ºF). Suppose a 90% confidence interval is constructed instead using the percentile method. Which of the following is a legitimate lower bound? a) 98.02ºF b) 98.26ºF c) 98.14ºF

b) 98.26ºF - A 90% confidence interval will be narrower than a 95% confidence interval. This implies that the lower bound for the 90% confidence interval will have to be greater than the lower bound of the 95% confidence interval.

Which of the following options best describes the values that make up a bootstrap distribution? a) A bootstrap distribution is a list of many, many sample statistics b) A bootstrap distribution is a list of many, many bootstrap sample statistics c) A bootstrap distribution is a list of many, many population parameters

b) A bootstrap distribution is a list of many, many bootstrap sample statistics

In which one of the following scenarios is it not appropriate to use the two-sample t-methods to analyze the data? a) Some manufacturers claim that non-hybrid sedan cars have a lower mean miles-per-gallon (mpg) than hybrid ones. Suppose that consumers randomly test 21 hybrid sedans and 31 non-hybrid sedans to test this hypothesis. It is believed that the miles-per-gallon for each type of sedan is normally distributed. b) An experiment is conducted to show that blood pressure can be consciously reduced in people trained in a biofeedback exercise program. Six subjects were randomly selected and blood pressure measurements were recorded before and after the training to answer this question of interest. c) A group of transfer bound students wondered if they will spend the same mean amount on textbooks and supplies each year at their four-year university as they have at their community college. They conducted a random survey of 54 students at their community college and 66 students at their local four-year university.

b) An experiment is conducted to show that blood pressure can be consciously reduced in people trained in a biofeedback exercise program. Six subjects were randomly selected and blood pressure measurements were recorded before and after the training to answer this question of interest.

Why is a bootstrap distribution used? a) To obtain an estimate of the population distribution in order to calculate the standard error of a statistic. b) To obtain an estimate of the sampling distribution in order to calculate the standard error of a statistic. c) To obtain an estimate of the sampling distribution in order to calculate the standard deviation of the sample data.

b) To obtain an estimate of the sampling distribution in order to calculate the standard error of a statistic.

Suppose α = 0.05, power = 0.85, and p-value = 0.3 What type of error could have been made? a) Type I Error b) Type II Error

b) Type II Error • Since the p-value > α, the decision is to fail to reject H0

A deck of playing cards has 52 cards, divided into four suits (diamonds, spades, clubs, and hearts). There are an equal number of cards in each suit in this deck. If three cards are drawn, what is the probability that the first two cards drawn are clubs and the last is red? There are three events in this problem: Let event A = first card drawn is a club Let event B = second card drawn is a club C = third card drawn is red. We want to find P(A and B and C) (i.e. the probability that the first card drawn is a club AND the second card drawn is a club AND the third card drawn is a red card). Under what condition will these three events be independent of each other? a) Never as the probability that the third card drawn is a red will depend on the color of the first two cards drawn. b) As long as the cards are sampled with replacement, the three events will be independent. c) As long as the cards are sampled without replacement, the three events will be independent. d) Regardless of how the cards are sampled, the events will be independent because what is drawn on any one draw will not be affected by what was drawn on any previous draw.

b) As long as the cards are sampled with replacement, the three events will be independent.

Suppose that a hypothesis test is conducted at the 5% significance level and that a p-value of 0.09 is the result. What is the conclusion of the test? a) Reject the null b) Fail to reject the null c) Accept the alternative d) Accept the null

b) Fail to reject the null

Suppose α = 0.10. With a p-value of 0.1113, what decision would be made? a) Reject the null hypothesis. b) Fail to reject the null hypothesis c) Accept the null hypothesis. d) Accept the alternative hypothesis as true. e) Reject the null hypothesis and, therefore, accept the alternative hypothesis as true f) Fail to reject the null hypothesis and, therefore, accept the null hypothesis as true.

b) Fail to reject the null hypothesis

Which of the following are possible conclusions from a hypothesis test using the significance level approach? (Select all that apply) a) fail to reject the alternative hypothesis b) Fail to reject the null hypothesis c) Accept the null hypothesis d) Reject the null hypothesis

b) Fail to reject the null hypothesis d) Reject the null hypothesis

Select all options below that show incorrect notation for a null hypothesis? (Select all that apply) a) H0: μ = 0 b) H0: μ > 0 c) H0: μ ≠ 0 d) H0: μ < 0

b) H0: μ > 0 c) H0: μ ≠ 0 d) H0: μ < 0 • A null hypothesis is always a statement that reflects the idea of no difference, no effect, or nothing has changed

b) H0: μns − μs≠0 • This notation conveys the idea that there is some kind of different between the two mean birth weights, but we do not know the direction of this difference. That is why the symbol ≠ is used.

Suppose researchers trying to determine if the mean diameter of trees at breast height for trees in previously clearcut areas was smaller than that mean diameter of trees at breast height for trees of similar age in non-clearcut areas. This resulted in a p-value of 0.3677. At a 10% significance level, could we make the decision that the null hypothesis is true? a) Yes b) No

b) No • We never make a decision to accept a null hypothesis! In this example, since the 0.3677 > 0.10, our decision is to NOT say the alternative hypothesis is true. That is, we would "fail to reject the null hypothesis"

Suppose researchers want to test that there is no difference in the mean heights of American males and females. They randomly sample 1000 American males and 1000 American females and calculate their heights as well as obtain sample averages. With these averages, they perform a hypothesis test which results in a p-value of 0.023 and they conclude that the null hypothesis of no difference in means can be rejected. A few months after publishing an article on their findings, they find another article from the Census Bureau which collects population data, stating that the true difference in the population average heights between American males and American females is much greater than 0. What type of error did the researchers make, if any? a) Type II error b) No error c) Type I error

b) No error • The researchers rejected the null hypothesis of no difference and it turns out that this was a the decision they should have made. The truth, according to the Census Bureau, is that the difference is greater than 0.

Are the bounds of a 95% confidence interval from the sample of 100 the same as the bounds of the 95% confidence interval from the sample of 1000? a) Yes, although it has nothing to do with the sample proportions being the same. Regardless of what the sample proportions are, the width of a confidence interval is not affected by the sample size. b) No. The confidence interval from the sample size of 100 is wider than the confidence interval from the sample size of 1000. c) No. The confidence interval from the sample size of 1000 is wider than the confidence interval from the sample size of 100. d) Yes since the sample proportion is the same in both samples.

b) No. The confidence interval from the sample size of 100 is wider than the confidence interval from the sample size of 1000.

8. Suppose x̅ = 25, s = 6.7, n = 35, and μo = 28. A one-sample t-test was performed using the hypotheses Ho: μ = μo and HA: μ < μo. The t-statistic was found to be -2.65. Which of the following is the correct interpretation of this t-statistic? a. 25 is 2.65 units below 28. b. The probability of observing a sample mean of 25 is 2.65 less likely than observing a null hypothesized value of 28. c. 25 is 2.65 standard errors below the null hypothesized value of 28. d. It is impossible to get a negative t-statistic so something must have been miscalculated.

c. 25 is 2.65 standard errors below the null hypothesized value of 28.

Suppose that a New Zealand based bio-pharmaceutical company wants to determine whether they should begin research into a new drug concerning hand joint pain and hair braiding. They randomly contact 932 hair braiding salons in Nigeria, Ghana, and Cameroon and ask salon workers if they have joint paint. They find that the proportion of salon workers in their sample who answered yes to the question about joint pain is 0.16. They use a bootstrap test of proportions to test the hypothesis that the proportion is greater than 0.15 at the 5% significance level, using the sample data, from which they obtain a p-value of 0.001. Is there practical significance, statistical significance or neither? a) No practical significance, no statistical significance b) Statistical significance, no practical significance c) Practical significance, no statistical significance

b) Statistical significance, no practical significance • Since the p-value was less than 0.05 there is strong evidence for the alternative hypothesis which indicates that there is statistical significance. However, consider the fact that the test was for the population proportion of hair braiders who have joint pain being greater than 0.15. If 15% of people in the population report that they have joint pain is this significant enough to begin research into a new drug concerning hand joint pain and hair braiding. That is, is it practical for the company to begin spending time and money pursuing this research path if only 15% of the population make up their possible clientele?

Sleep deprivation continues to be widespread in America. According to a National Sleep Foundation poll, a majority of American adults (63%) do not get the recommended eight hours of sleep needed for good health, safety, and optimum performance. In fact, nearly one-third (31%) report sleeping less than seven hours each week night, though many adults say they try to sleep more on weekends. How can a researcher "make" the data statistically significant? a) Use more bootstrap samples in the bootstrap distribution b) Take a larger sample size c) Do not determine the hypotheses until after data are collected d) Take samples with very different sample means e) A researcher can't make data statistically significant.

b) Take a larger sample size

A university chancellor is interested in the average number of miles biked each week by students on campus. They employ the statistics department of the university to randomly sample 1345 students and collect data on how many miles they bike each week. The chancellor believes that the average is less than zero miles and wants to perform a hypothesis test based on this opinion despite advice from the statistics department that her hypothesis is not possible. The statistics department goes forward with the hypothesis test and finds a sample mean 15 miles and obtains a p-value of 0.97 What best explains the reason for the p-value of 1? a) Every student on campus bikes at her school. b) The chancellor decided on an alternative hypothesis that was unrealistic one-sided alternative hypothesis. In fact, the chancellor picked the wrong side and therefore the no evidence to support the alternative that the mean number of miles biked each week by students on campus was less than 0. c) None of the students on campus bike at her school.

b) The chancellor decided on an alternative hypothesis that was unrealistic one-sided alternative hypothesis. In fact, the chancellor picked the wrong side and therefore the no evidence to support the alternative that the mean number of miles biked each week by students on campus was less than 0.

Data were collected in North Carolina over the period of a year on 150 randomly selected mothers and their newborn babies. Although many variables were measured, researchers were interested in whether or not the mother was a smoker and the weight of their newborn baby. Researchers would like to know if there was evidence that newborns from mothers who smoke have a different average birth weight than newborns from mothers who don't smoke. The following summary statistics were found on each group of newborns. Consider the samples of newborns as representative of all newborns born to smoking and nonsmoking mothers. Smoker mean 6.78 standard deviation 1.43 sample size 50 Nonsmoker mean 7.18 standard deviation 1.6 sample size 100 Do the data provide evidence of a difference in birth weight for babies born to mothers who smoke and mothers who do not? Use a significance level = 0.05. Which of the following is the correct null hypothesis in words? a) The mean birth weight of babies born to mothers who do not smoke is different than the mean birth weight of babies born to mothers who do smoke. b) The mean birth weight of babies born to mothers who do not smoke is the same as the mean birth weight of babies born to mothers who do smoke. c) The mean birth weight of babies born to mothers who do not smoke is less than the mean birth weight of babies born to mothers who do smoke. d) The mean birth weight of babies born to mothers who do not smoke is greater than the mean birth weight of babies born to mothers who do smoke.

b) The mean birth weight of babies born to mothers who do not smoke is the same as the mean birth weight of babies born to mothers who do smoke. • The null hypothesis is usually one that sounds like "no difference in population means" especially for a two sample problem like this one.

Five students visiting the student health center for a free dental examination during National Dental Hygiene Month were asked how many months had passed since their last visit to a dentist. Their responses (in months) were 6, 17, 11, 22, and 29. A bootstrap distribution from 10,000 bootstrap sample means was generated. original sample: min 6 Q1 11 median 17 Q3 22 max 29 mean 17 sd 9.027735 n 5 missing 0 bootstrap distribution: min 6 Q1 14.6 median 17 Q3 19.4 max 29 mean 16.98124 sd 3.616919 n 10000 missing 0 The bootstrap distribution of the histogram is slightly right skewed. Why? a) Not enough bootstrap samples were generated. If we generated more (say 20,000), the bootstrap distribution would look more like a normal curve. b) The sample size (n = 5) is small. If a larger sample size (such as n = 30) was taken, the bootstrap distribution would have a more normal shape curve as long as the population data were not too skewed. c) The widths of the bins in the histogram are too wide. If the bin width was 1 month instead of 2 months, the bootstrap distribution would look more like a normal curve. d) All three of the above play a role in why the bootstrap distribution is slightly right skewed. e) Only the first two choices above play a role in why the bootstrap distribution is slightly right skewed.

b) The sample size (n = 5) is small. If a larger sample size (such as n = 30) was taken, the bootstrap distribution would have a more normal shape curve as long as the population data were not too skewed.

If a decision is made to reject the null hypothesis a) A Type I Error was definitely made. b) There is a chance that a Type I Error was made. c) There is no chance that a Type I Error was made.

b) There is a chance that a Type I Error was made.

Suppose there are 5 people in a room and we are interested in the average shoe size of people in the room. We randomly sample 2 people from the room and take the average of their shoe sizes. We repeat this process again and again until every possible unique combination of two people has been used and we take the values and construct a sampling distribution. How many sample statistics will the sampling distribution have? Keep in mind that only unique samples are used to calculate the sample statistic. For example if we randomly select person 1 and person 2 and then randomly select person 2 and person 1 these two samples are not unique because they consist of the same two individuals. Also, the same person cannot be selected twice to make up an entire sample. a) 25 b) 15 c) 10

c) 10 - We can begin by assigning each person a number and then determining how many different people they can be matched with. The cells correspond to the two people that will make up the sample. Without any restrictions there are 25 possible combinations. Cell (1, 1) is red because we cannot have the same person repeated twice in a sample. The same is true for cell (2, 2), cell (3, 3), and so on. The greyed out cells are not counted because having person 1 and person 2 (1, 2) is the same as having person 2 and then person 1 (2, 1) so the samples beneath the diagonal are the same as the samples above the diagonal. Upon counting the number of white cells we see that there are 10 possible combinations.

c) 100 • Since we use the bootstrap distribution to calculate a p-value we need it to be centered at the null value because a p-value is the probability of observing what was observed or something more extreme than the one we observed if the null hypothesis is true. Since ODOT believes that the mean is 100, this is the population mean under the null and, therefore, the bootstrap distribution should be centered at 100.

Suppose that the following conclusion is made from a hypothesis test: With 95% confidence, the average height of a Tanzanian man is estimated to be between 70.35 inches and 79.21 inches. Which of the following null hypothesized values for the population mean would rejected, based on this 95% confidence interval. (Check all that apply) a) 72 inches b) 79.21 inches c) 69 inches d) 50.4 inches

c) 69 inches d) 50.4 inches

Which confidence interval is widest? a) 90% confidence interval b) 95% confidence interval c) 99% confidence interval

c) 99% confidence interval

Suppose that in a jury trial, the jury gives the judge the verdict "not guilty." Suppose that in a different trial, the jury made a decision where a Type I Error has a chance of occurring. Which of the following statements is true about making a Type I Error in a jury trial? a) A Type I Error means the jury gave the judge a verdict of "not guilty" when the defendant is indeed innocent. b) A Type I Error means the jury gave the judge a verdict of "guilty" when the defendant is indeed guilty. c) A Type I Error means the jury gave the judge a verdict of "guilty" when in fact the defendant is innocent. d) A Type I Error means the jury gave the judge a verdict of "not guilty" when in fact the defendant is guilty.

c) A Type I Error means the jury gave the judge a verdict of "guilty" when in fact the defendant is innocent. • Making a Type I Error means saying that an innocent person is guilty (and possibly sending an innocent person to jail)

Which of the following is the correct definition of a bootstrap statistic? a) A bootstrap statistic is a statistic calculated from the original sample. b) A bootstrap statistic is a population parameter calculated from the population. c) A bootstrap statistic is the statistic calculated from the bootstrap sample.

c) A bootstrap statistic is the statistic calculated from the bootstrap sample.

One card is drawn from a deck of playing cards. Let A = {card drawn is a club}. What is the complement of event A a) A^c = the card drawn is the ace of diamonds b) A^c = the card drawn is red c) A^c = the card drawn is not a club d) A^c = the card drawn is a spade

c) A^c = the card drawn is not a club

c) Amount expected to be spent on holiday gifts this year

Chilean miners often find themselves underground for many hours on end. A mining company reports a sample of 30 miners spend on average 9.4 hours underground per day. It is estimated with 95% confidence that the average time spent underground by all minors at this company is between 8 hours to 10.8 hours per day. What is the population parameter that researchers are trying to estimate? a) Average number of hours miners work per day. b) Average number of hours spent underground per day by all miners c) Average number of hours spent underground per day by Chilean miners in this particular mining company

c) Average number of hours spent underground per day by Chilean miners in this particular mining company

Consider the random experiment of rolling one six-sided dice. Each side contains one number from 1 to 6 such that only one number appears on the dice. What are the possible outcomes on one trial? a) the number of times each value is rolled b) the number of times a certain value is rolled c) the number of times the die is rolled until a certain value (such as a 1) appears d) the value that is rolled (1, 2, 3, 4, 5, or 6) e) whether or not a certain value (such as a 1) is rolled

d) the value that is rolled (1, 2, 3, 4, 5, or 6)

The Census Bureau randomly samples 340 college freshmen in America and 259 college freshmen in Brazil and asks them how many credit hours they are taking. They find that the difference in the average number of credit hours being taken by students in each country is 0.1 credit hours. In other words, they find that the average number of credit hours taken by American students per term is 0.1 more than the average number of credit hours taken by students in Brazil. They perform a hypothesis test on the data at the 5% significance level and obtain a p-value of 0.01. Select the correct option that describes the practical and statistical significance of this study. a) There is no practical significance because a difference in means of 0.1 is not very much, considering the data was credit hours. Because the p-value was so small, there is also no statistical significance. b) There is neither practical or statistical significance present in the study. c) Because the p-value was so small, the results are statistically significant. There is no practical significance because a difference in means of 0.1 is not very much, considering the data was credit hours. d) There is both practical and statistical significance present in the study.

c) Because the p-value was so small, the results are statistically significant. There is no practical significance because a difference in means of 0.1 is not very much, considering the data was credit hours. • We get statistical significance because the p-value was less than the significance level, but the difference we observe was very small which leads to no real practical significance.

If typical probabilities of making a Type II Error are between 0.1 and 0.2, what are typical levels of power? a) Between 0.1 and 0.2 b) The complement of the significance level c) Between 0.8 and 0.9 d) The significance level

c) Between 0.8 and 0.9

Suppose α = 0.05, power = 0.85, and p-value = 0.3 What decision is made about the null hypothesis? a) Accept the null hypothesis b) Reject the null hypothesis c) Fail to reject the null hypothesis

c) Fail to reject the null hypothesis • Since the p-value > α, the decision is to f• Since the p-value > α, the decision is to fail to reject H0ail to reject H0

Data were collected in North Carolina over the period of a year on 150 randomly selected mothers and their newborn babies. Although many variables were measured, researchers were interested in whether or not the mother was a smoker and the weight of their newborn baby. Researchers would like to know if there was evidence that newborns from mothers who smoke have a different average birth weight than newborns from mothers who don't smoke. The following summary statistics were found on each group of newborns. Consider the samples of newborns as representative of all newborns born to smoking and nonsmoking mothers. Smoker mean 6.78 standard deviation 1.43 sample size 50 Nonsmoker mean 7.18 standard deviation 1.6 sample size 100 Do the data provide evidence of a difference in birth weight for babies born to mothers who smoke and mothers who do not? Use a significance level = 0.05. Which of the following is the correct null hypothesis in statistical notation? μns represents the mean birth weight (in pounds) for all babies born to mothers who do not smoke and μs is the mean birth weight (in pounds) for all babies born to mothers who do smoke. a) H0: μns − μs<0 b) H0: μns − μs≠0 c) H0: μns − μs=0 d) H0: μns − μs>0

c) H0: μns − μs=0 • This notation conveys the idea that there is no difference between the two mean birth weights.

An unbalanced coin is tossed two times. On this coin, the probability of getting a heads to land face up on any one toss is 0.4. There are four possible outcomes in the sample space (HH, HT, TH, TT), where HT (for example) means heads on first toss AND tails on second toss. One method to find P(at least one head) is to apply the addition rule. How can the complement rule be used to find this probability? a) P(at least one head) = 1 - P(exactly one head) b) P(at least one head) = 1 - P(one head or less) c) P(at least one head) = 1 - P(no heads) d) P(at least one head) = 1 - P(both heads)

c) P(at least one head) = 1 - P(no heads, .36) = 0.64.

Consider the random experiment of selecting an envelope from a prize basket while blindfolded. The prize can either be monetary (like an envelope with a check) or a take-away prize (like an envelope with the title to a new car). Using the complement rule, what is the probability that at least one of the envelopes contains a take away prize? a) P(at least one take-away prize) = 1 - P(one take-away prize) b) P(one take-away prize) = P(at least one take-away prize) c) P(at least one take-away prize) = 1 - P(no take-away prize)

c) P(at least one take-away prize) = 1 - P(no take-away prize)

The Oregon Department of Transportation is interested in the average number of miles driven by working adults aged 35 - 46 in a one week period. ODOT believes that working adults in this age range drive more than 100 miles in a one week period. ODOT sends a survey to 1000 such adults and surveys are returned by 274 adults. The survey asks respondents for an estimate of how many hours they believe they drive each week. ODOT determines that the average number of miles driven each week by the respondents of the survey is 74.9 miles. Use the correct p-value (.0154) from the last question to make a decision and write a conclusion. What decision would be made at α = 0.05? a) Fail to reject the null hypothesis b) Accept the null hypothesis c) Reject the null hypothesis

c) Reject the null hypothesis

Which of the following most closely resembles the goal of simulating a random experiment? a) So that we can use technology that is complicated to control and implement b) So that we can repeat the experiment under different conditions over time c) So that the experiment can be easily replicated many times quickly

c) So that the experiment can be easily replicated many times quickly

Sleep deprivation continues to be widespread in America. According to a National Sleep Foundation poll, a majority of American adults (63%) do not get the recommended eight hours of sleep needed for good health, safety, and optimum performance. In fact, nearly one-third (31%) report sleeping less than seven hours each week night, though many adults say they try to sleep more on weekends. Even though everything was the same between the four studies (same sample mean and same sample standard deviation), different decisions were made. Why were different decisions made? a) Just by the luck of how the bootstrap distributions were created b) An error was made with the calculation of the p-values. If the sample means and sample standard deviations were the same, the p-values should be the same no matter what the sample size is c) The sample sizes were different for the four different studies d) The bootstrap distributions were generated with a different number of bootstrap sample means

c) The sample sizes were different for the four different studies • The sample means and the sample standard deviations were the same in all four studies. The only thing that was different was the sample size

Five students visiting the student health center for a free dental examination during National Dental Hygiene Month were asked how many months had passed since their last visit to a dentist. Their responses (in months) were 6, 17, 11, 22, and 29. A bootstrap distribution from 10,000 bootstrap sample means was generated. original sample: min 6 Q1 11 median 17 Q3 22 max 29 mean 17 sd 9.027735 n 5 missing 0 bootstrap distribution: min 6 Q1 14.6 median 17 Q3 19.4 max 29 mean 16.98124 sd 3.616919 n 10000 missing 0 Compare the summary statistics of the original sample data to the histogram and summary statistics from the bootstrap distribution. Which of the following statements is true? a) The spread of the original sample and bootstrap distribution is roughly the same, but the center of the bootstrap distribution is higher than the center of the original sample data. b) The center of the original sample and bootstrap distribution is roughly the same, but there is more spread in the bootstrap distribution than the in the original sample data. c) The center of the original sample and bootstrap distribution is roughly the same, but there is less spread in the bootstrap distribution than the in the original sample data. d) The spread of the original sample and bootstrap distribution is roughly the same, but the center of the bootstrap distribution is lower than the center of the original sample data. e) The center and spread of the original sample and bootstrap distribution are roughly the same.

c) The center of the original sample and bootstrap distribution is roughly the same, but there is less spread in the bootstrap distribution than the in the original sample data.

James has a deck of 52 playing cards. A fair deck of cards has an equal number of red and black cards (26 of each). James's friend, Grace, wonders if the deck is unfair, meaning there are more of one color than the other. To determine if the deck is not fair, cards are drawn one at a time with replacement. Which of the following is the correct null hypothesis in words? a) The deck is not fair. b) There is no evidence that the deck is fair. c) The deck is fair. d) There is strong evidence that the deck is fair.

c) The deck is fair.

Sleep deprivation continues to be widespread in America. According to a National Sleep Foundation poll, a majority of American adults (63%) do not get the recommended eight hours of sleep needed for good health, safety, and optimum performance. In fact, nearly one-third (31%) report sleeping less than seven hours each week night, though many adults say they try to sleep more on weekends. Which of the following explains why the data may not be practically significant? a) The sample means for all four studies were the same. b) The sample mean is less than 8 hours when the alternative hypothesis was that the mean is greater than 8 hours. c) The difference between the sample mean (7.9 hours/night) and the hypothesized value of the population mean (8 hours) is only 0.1 hour (6 minutes), which is not very different - the sample mean and hypothesized value are nearly identical. d) The sample size for the last study was extremely large. e) The sample size for the first study was extremely small.

c) The difference between the sample mean (7.9 hours/night) and the hypothesized value of the population mean (8 hours) is only 0.1 hour (6 minutes), which is not very different - the sample mean and hypothesized value are nearly identical.

Suppose the US Department of Education is interested in the difference in the average entry-level salary of students who graduate from college with a degree in a core science subject and those who graduate from college with a degree in a core arts subject. They collect data from 100 universities in the US on the salaries of their most recent graduates. What is the parameter of interest? a) The difference in the median entry-level salary of graduates from a core science subject and those from a core art subject. b) The mean entry-level salary of graduates from college. c) The difference in the average entry-level salary of all graduates from a core science subject and those from a core art subject.

c) The difference in the average entry-level salary of all graduates from a core science subject and those from a core art subject.

Data were collected in North Carolina over the period of a year on 150 randomly selected mothers and their newborn babies. Although many variables were measured, researchers were interested in whether or not the mother was a smoker and the weight of their newborn baby. Researchers would like to know if there was evidence that newborns from mothers who smoke have a different average birth weight than newborns from mothers who don't smoke. The following summary statistics were found on each group of newborns. Consider the samples of newborns as representative of all newborns born to smoking and nonsmoking mothers. Smoker mean 6.78 standard deviation 1.43 sample size 50 Nonsmoker mean 7.18 standard deviation 1.6 sample size 100 Do the data provide evidence of a difference in birth weight for babies born to mothers who smoke and mothers who do not? Use a significance level = 0.05. Which of the following is the correct alternative hypothesis in words? a) The mean birth weight of babies born to mothers who do not smoke is less than the mean birth weight of babies born to mothers who do smoke. b) The mean birth weight of babies born to mothers who do not smoke is greater than the mean birth weight of babies born to mothers who do smoke. c) The mean birth weight of babies born to mothers who do not smoke is different than the mean birth weight of babies born to mothers who do smoke. d) The mean birth weight of babies born to mothers who do not smoke is the same as the mean birth weight of babies born to mothers who do smoke.

c) The mean birth weight of babies born to mothers who do not smoke is different than the mean birth weight of babies born to mothers who do smoke. • The question of interest asks, "Do the data provide evidence of a difference in birth weight for babies born to mothers who smoke and mothers who do not?" The alternative hypothesis is associated with the question of interest.

What does it mean for a confidence interval to "capture" the population parameter? a) All values in the confidence interval equals the population parameter b) The value that was the population parameter no longer equals the population parameter as it being captured by the confidence interval means it is absorbed into the confidence interval as just another value c) The population parameter is one of the values in the range of values of the confidence interval d) The population parameter is not one of the values in the range of values of the confidence interval

c) The population parameter is one of the values in the range of values of the confidence interval

Consider the random experiment of rolling one six-sided dice. Each side contains one number from 1 to 6 such that only one number appears on the dice. What is the trial? a) what lands face-up on one roll of the dice b) rolling the dice until a certain outcome appears c) one roll of the dice

c) one roll of the dice

We randomly sample 50 items from Target stores and note the price for each item. Then we visit Walmart and collect the price for each of those same 50 items. Which of the following is true? a) The sample sizes are not large enough to be able to say that the distribution of differences in sample means x̅1 − x̅2 is normally distributed b) The variable of interest is not quantitative, therefore the conditions of the two-sample t-methods have not been met. c) The scenario describes paired data, therefore the conditions of the two-sample t-methods have not been met. d) All of the conditions of the two-sample t-methods have been met in this scenario.

c) The scenario describes paired data, therefore the conditions of the two-sample t-methods have not been met. • The problem describes paired data because 50 items were selected from Target than those same 50 items were found and priced at Walmart. Since the data is pared, we cannot use two-sample t-methods on the data. Instead, we should use paired t-methods.

Coach S.P. Dee knows his team's runners of the 100-meter event will need to run faster than 12 seconds, on average, for their team to have a chance to repeat as conference champions at the end of the year. He records the times of the 100-meter event for his team's runners at three different track meets towards the end of the year. What is the variable of interest? What type of variable is it? a) The number of meets in which a sprinter on the team runs the 100-meter dash in under 12 seconds. It is quantitative. b) The length of the race. It is quantitative. c) The time to complete the 100-meter dash. It is quantitative. d) The number of meets in which a sprinter runs the 100-meter dash. It is quantitative. e) Whether or not a sprinter on the team runs the 100-meter dash in under 12 seconds. It is categorical.

c) The time to complete the 100-meter dash. It is quantitative. • The primary outcome for each sprinter is the time it takes them to run the 100-meter dash. Since it is a number that is being recorded, the variable is quantitative.

True or False? The distribution of the sample mean is symmetric and bell-shaped. a) This is always true regardless of the population distribution. b) This is true only if the sample size is larger than 30. c) This is true if the population that the sample came from is normal or if the sample size is sufficiently large.

c) This is true if the population that the sample came from is normal or if the sample size is sufficiently large.

A manufacturer of paper coffee cups would like to estimate the proportion of cups that are defective (tears, broken seems, etc.) from a large batch of cups. They take a random sample of 200 cups from the batch of a few thousand cups and found 18 to be defective. The goal is to perform a hypothesis test to determine if the proportion of defective cups made by this machine is more than 8%. Based on your answers to the previous problems, what is the conclusion to this hypothesis test? a) We reject the null hypothesis at the 0.05 significance level. The data provide no evidence to suggest that the proportion of defective cups made by this machine is more than 8%. b) We reject the null hypothesis at the 0.05 significance level. The data provide strong evidence to suggest that the proportion of defective cups made by this machine is more than 8%. c) We fail to reject the null hypothesis at the 0.05 significance level. The data provide no evidence to suggest that the proportion of defective cups made by this machine is more than 8%. d) We fail to reject the null hypothesis at the 0.05 significance level. The data provide strong evidence to suggest that the proportion of defective cups made by this machine is more than 8%.

c) We fail to reject the null hypothesis at the 0.05 significance level. The data provide no evidence to suggest that the proportion of defective cups made by this machine is more than 8%.

Here is the output for the bounds of a 95% confidence interval for μ using the percentile method: 2.5%, 775.20 97.5%, 909.98 Which of the following is a correct interpretation of the confidence interval? a) There's a 95% chance that shoppers at this mall expect to spend between $775.20 and $909.98 more on holiday gifts this year than last year. b) We're 95% confident that the all shoppers at this mall expect to spend between $775.20 and $909.98 more on holiday gifts this year than last year. c) We're 95% confident that the mean amount shoppers at this mall expect to spend on holiday gifts this year is between $775.20 and $909.98. d) We're 95% confident that the mean amount shoppers at this mall expect to spend on holiday gifts this year is between $775.20 and $909.98 more than the mean from last year. e) 95% of all shoppers at this mall expect to spend between $775.20 and $909.98 on holiday gifts this year.

c) We're 95% confident that the mean amount shoppers at this mall expect to spend on holiday gifts this year is between $775.20 and $909.98.

A hospital is trying to cut down on emergency room wait times. It is interested in the amount of time patients must wait before being called to be examined. An investigation committee randomly sampled 70 patients and recorded the wait time for each. The sample mean was 1.5 hours with a sample standard deviation of 0.55 hours. Does the data provide evidence that the mean wait time is less than 1.75 hours? Can t-methods be used to conduct a hypothesis for this problem? Select the best reason. a) Yes - the sample mean is larger than 1 and therefore the distribution of sample means is approximately normal by the Central Limit Theorem. b) Yes - the standard deviation greater than the 10% rule. c) Yes - the sample size is greater than 30, therefore we assume the distribution of sample means is approximately normal by the Central Limit Theorem. d) No - there is not enough information to determine if the t-methods could be used or not

c) Yes - the sample size is greater than 30, therefore we assume the distribution of sample means is approximately normal by the Central Limit Theorem. • The sample size is 70 in this example so we should feel pretty confident that the distribution of sample means is approximately normal. If the sampling distribution is normal, then t-methods are appropriate to use

In the jury trial analogy, "the defendant is not guilty" is which hypothesis? a) the null hypothesis b) the alternative hypothesis c) neither the null nor the alternative hypothesis

c) neither the null nor the alternative hypothesis • "Not guilty" does not imply the defendant is innocent. In the jury trial analogy, it means that the jury is not convinced beyond a reasonable doubt that the defendant is guilty, so they report "not guilty".

An unbalanced coin is tossed two times. On this coin, the probability of getting a heads to land face up on any one toss is 0.4. There are four possible outcomes in the sample space (HH, HT, TH, TT), where HT (for example) means heads on first toss AND tails on second toss. What is the complement of "at least one head" in two tosses of the coin? a) exactly one head was tossed in the two tosses b) both tosses were heads c) neither toss was a head d) more than one of the above

c) neither toss was a head

A study is designed to determine whether grades in a statistics course could be improved by offering special review material. The 250 students enrolled in a large introductory statistics class are also enrolled in one of 20 lab sections. The 20 lab sections are randomly divided into 2 groups of 10 lab sections each. The students in the first set of 10 lab sections are given extra review material during the last 15 minutes of each weekly lab session. The students in the remaining 10 lab sections receive the regular lesson material, without the extra review material. The final grades of the students who reviewed weekly were higher, on average, than the students who did not review every week. 2. Suppose the instructor performed two-sample t-test for the difference in means and found a one-sided p-value of 0.02. What is the best interpretation of this p-value? a. If the review material really does help improve grades of students, 2% of all random samples would result in a difference in sample means more than the observed difference. b. If the review material does help improve grades of students, 98% of all samples would result in a difference in sample means more than the observed difference. c. If the review material truly did not help improve grades of students, 2% of all random samples would result in a difference in sample means more than the observed difference. d. If the review material truly did not help improve grades of students, 98% of all random samples would result in a difference in sample means similar to what was observed.

c. If the review material truly did not help improve grades of students, 2% of all random samples would result in a difference in sample means more than the observed difference.

Two events are ______ if they are the only two events in the sample space AND they cannot occur at the same time.

complements

A ______ ______ can be used to make a decision about a null hypothesis, but only if the alternative is "not equal to" (i.e. a two-sided hypothesis test).

confidence interval

Suppose α = 0.10, β = 0.13, and p-value = 0.03 What is the power of the test? a) .03 b) .10 c) .13 d) .87 e) .90 f) .97

d) .87 • Power = 1 - P(Type II Error) = 1 - β = 1 - .13 = .87

Researchers wondered if a new reading program would have an effect on test scores among 4th through 6th graders. In a particular large school district, they randomly chose half the 4th through 6th grade classes and assigned the teachers in those classes to teach using the new reading program. The remaining 4th through 6th grade classes would continue to use the existing reading program. At the end of the year, a standard exam was given to all 4th through 6th graders and the average score on the exam was compared between those who used the new reading program and those who used the existing reading program. What are the two groups being compared? a) 4th through 6th graders in this large school district who performed better using the new reading program and 4th through 6th graders in this large school district who performed worse using the existing reading program. b) 4th through 6th graders in this large school district and 4th through 6th graders not in this large school district c) 4th graders and 6th graders d) 4th through 6th graders in this large school district who used the new reading program and 4th through 6th graders in this large school district who did used the existing reading program. e) 4th through 6th graders in this large school district and other students in this large school district.

d) 4th through 6th graders in this large school district who used the new reading program and 4th through 6th graders in this large school district who did used the existing reading program.

Which one of the following is a condition for the two-sample t-methods? a) A random sample of cases is taken from two independent populations. b) Both sample sizes are "large" and/or the data in each sample are symmetric or only slightly skewed c) The cases/individuals in two groups are independent of each other. d) All of the above

d) All of the above

An interval estimate is a list of values we believe the parameter will be. When is an interval estimate called a confidence interval? a) An interval estimate is called a confidence interval when we know the population parameter b) An interval estimate is called a confidence interval when we know the sample statistic c) An interval estimate is called a confidence interval whenever we list a range of possible values for the parameter d) An interval estimate is called a confidence interval when a level of confidence is given that one of the values in the interval is the value of the parameter e) An interval estimate is never called a confidence interval

d) An interval estimate is called a confidence interval when a level of confidence is given that one of the values in the interval is the value of the parameter

Researchers wondered if a new reading program would have an effect on test scores among 4th through 6th graders. In a particular large school district, they randomly chose half the 4th through 6th grade classes and assigned the teachers in those classes to teach using the new reading program. The remaining 4th through 6th grade classes would continue to use the existing reading program. At the end of the year, a standard exam was given to all 4th through 6th graders and the average score on the exam was compared between those who used the new reading program and those who used the existing reading program. What is the variable of interest? a) Whether or not there is a difference in the average exam scores b) Between classes using the new reading program and classes using the existing reading program c) Which reading program a student's class is using d) End-of-year standard exam score e) The number of students using each reading program grade level of student

d) End-of-year standard exam score

Coach S.P. Dee knows his team's runners of the 100-meter event will need to run faster than 12 seconds, on average, for their team to have a chance to repeat as conference champions at the end of the year. He records the times of the 100-meter event for his team's runners at three different track meets towards the end of the year. In words, which of the following is the correct null hypothesis? a) H0: all sprinters of the 100-meter dash on Coach Dee's team will take less than 12 seconds to finish the event. b) H0: the sprinters of the 100-meter dash on Coach Dee's team will take less than 12 seconds to finish the event, on average. c) H0: all sprinters of the 100-meter dash on Coach Dee's team will take 12 seconds to finish the event. d) H0: the sprinters of the 100-meter dash on Coach Dee's team will take 12 seconds to finish the event, on average. e) H0: the sprinters of the 100-meter dash on Coach Dee's team will take more than 12 seconds to finish the event, on average.

d) H0: the sprinters of the 100-meter dash on Coach Dee's team will take 12 seconds to finish the event, on average. • The null hypothesis has to indicate "equality" (i.e. contain an = if written in notation) AND include "average" or "mean" (if the variable of interest is quantitative).

Sleep deprivation continues to be widespread in America. According to a National Sleep Foundation poll, a majority of American adults (63%) do not get the recommended eight hours of sleep needed for good health, safety, and optimum performance. In fact, nearly one-third (31%) report sleeping less than seven hours each week night, though many adults say they try to sleep more on weekends. Let μ = the mean number of hours of sleep per night for all college students. Which of the following is the correct null and alternative hypotheses for all four studies? a) H0: μ = 7.9 hours, HA: μ < 7.9 hours b) H0: μ = 7.9 hours, HA: μ > 7.9 hours c) H0: μ = 8 hours, HA: μ = 7.9 hours d) H0: μ = 8 hours, HA: μ < 8 hours e) H0: μ = 8 hours, HA: μ > 8 hours

d) H0: μ = 8 hours, HA: μ < 8 hours • If the amount of sleep college students get per night, on average, is no different than the recommended amount, the average for all college students would be 8 hours - this is the null hypothesis. The question is wondering if college students get less than the recommended amount, which would be less than 8 hours, on average - this is the alternative hypothesis.

Store owners in a mall were thinking ahead to the holiday season. They were wondering if they would need to boost their inventory for this coming season compared to the previous holiday season. During the previous season, consumers spent an average of $800 on gifts. The store owners are wondering if consumers at their mall will spend more than $800, on average, on gifts this coming year. By walking around the mall during busier times in October, interviewers sampled 40 shoppers in the mall who expected to purchase gifts this coming year. In words, which of the following is the correct alternative hypothesis? a) HA: all shoppers at this mall expect to spend more than $800 on holiday gifts this year. b) HA: There is strong evidence to indicate that, among shoppers at this mall, the average amount expected to be spent on holiday gifts this year is more than $800. c) HA: For shoppers at this mall, the average amount expected to be spent on holiday gifts this year is $800. d) HA: For shoppers at this mall, the average amount expected to be spent on holiday gifts this year is more than $800. e) HA: more than half of shoppers at this mall expect to spend more than $800 this year on holiday gifts.

d) HA: For shoppers at this mall, the average amount expected to be spent on holiday gifts this year is more than $800. HA: µ > $800

Suppose that researchers are interested in the average number of cars sold in the US during 2017 Christmas time because they believe that the average number of cars sold during December 2017 was higher than the average number of cars sold during July 2017. They collect information from 100 randomly selected dealerships across the US on how many cars were sold in July 2017 and how many cars were sold in December 2017. They perform a hypothesis test on the data and determine that the p-value is 0.043. Which of the following is the best interpretation of the p-value? a) The average number of cars sold in December is 4.3% more compared to July. b) 4.3% of cars sold are either in December or July. c) The probability of observing two months with a large number of cars sold is 4.3%. d) If it were true that the mean number of cars sold in December was the same as in July, there is a 4.3% chance of observing a sample mean number of cars sold in December 2017 that is greater than the mean number of cars sold in July 2017.

d) If it were true that the mean number of cars sold in December was the same as in July, there is a 4.3% chance of observing a sample mean number of cars sold in December 2017 that is greater than the mean number of cars sold in July 2017.

20% of candies in an M&M's bag of milk chocolate candies are supposed to be orange. A student tested this claim by randomly sampling many bags and provided the following 95% confidence interval (using the formula method) for the proportion of orange candies: (0.16, 0.22). Based on the confidence interval, is there evidence to reject the claim that 20% of the candies in a bag of M&Ms milk chocolate are orange? a) Yes since 0.20 is not between the bounds of the confidence interval. b) Yes since the sample proportion is between the bounds of the confidence interval. c) No since 0.20 is not between the bounds of the confidence interval. d) No since 0.20 is between the bounds of the confidence interval. e) Yes since 0.20 is between the bounds of the confidence interval.

d) No since 0.20 is between the bounds of the confidence interval.

Suppose a p-value from a hypothesis test was 0.025. What decision would be made if the significance level is 0.05? a) Do not reject the claim in the null hypothesis. Therefore, do not say the claim in the alternative hypothesis is true. b) Do not say the claim in the alternative hypothesis is true. Therefore, accept the claim in the null hypothesis as true. c) Reject the claim in the null hypothesis but do not say the claim in the alternative hypothesis is true. d) Reject the claim in the null hypothesis and accept the claim in the alternative hypothesis.

d) Reject the claim in the null hypothesis and accept the claim in the alternative hypothesis. • Since the p-value of 0.025 < 0.05, the decision is to reject the claim in the null hypothesis FOR the claim in the alternative hypothesis, meaning accepting the claim in the alternative hypothesis as true.

James has a deck of 52 playing cards. A fair deck of cards has an equal number of red and black cards (26 of each). James's friend, Grace, wonders if the deck is unfair, meaning there are more of one color than the other. To determine if the deck is not fair, cards are drawn one at a time with replacement. What is the variable of interest? a) The total number of cards drawn before a red card is drawn b) Whether the deck is fair or not c) The total number of red cards in the deck d) The color of a card drawn e) The number of red cards drawn

d) The color of a card drawn • The piece of information being recorded on each case (i.e. each card) is the color of the card (or whether or not a red card is drawn).

Which of the following is NOT one of the decisions that can be made in a hypothesis test? a) The alternative hypothesis is true. b) The alternative hypothesis is not true c) The null hypothesis is not true d) The null hypothesis is true

d) The null hypothesis is true • We NEVER will accept the null hypothesis as true from a hypothesis test. As a reminder, a jury trial NEVER tells the judge that the defendant is innocent

Below are the summary statistics and histogram: min 355, Q1 715.75, median 862, Q3 999, max 1239, mean 842.625, sd 220.1796, n 40, missing 0 hypothesized value = 800 Using proper notation, what is the value of the sample mean? a) μ = $842.625 b) μ = $800 c) μx̅ = $842.625 d) x̅ = $842.625 e) x̅ = $800

d) x̅ = $842.625

18. A letter is randomly selected from the word MISSISSIPPI. What is the probability that the letter will be an s? a. 1/11 b. 3/10 c. 1/4 d. 4/11 e. 1/3

d. 4/11

A researcher wants to know if senior students drink more coffee than freshmen students. She takes a sample of 100 students from each of these populations and asks them how many cups of coffee they drink every day. The 95% confidence interval the difference between these two populations (seniors-freshmen) ranges from 2 to 5. Which one of the following choices is the correct interpretation of this interval? a) We are 95% confident that freshmen students, on average, drink 2 to 5 more cups of coffee than senior students on a daily basis. b) We are 95% confident that senior students in her sample drink 2 to 5 more cups of coffee than freshmen students in her sample on a daily basis, on average. c) With 95% probability, senior students drink 2 to 5 more cups of coffee than freshmen students on a daily basis. d) We are 95% confident that senior students, on average, drink 2 to 5 more cups of coffee than freshmen students on a daily basis.

d) We are 95% confident that senior students, on average, drink 2 to 5 more cups of coffee than freshmen students on a daily basis.

The equatorial radius of the planet Jupiter is measured 40 times independently by a process that is practically free of bias. The average of these measurements, x̅ = 71,492 kilometers. From a bootstrap distribution of several thousand samples, SEx̅ = 4.4 km. Suppose the 95% confidence interval was (71,462 km, 71,522 km). (Note: these are NOT the correct bounds!) Which of the following is a correct interpretation of this confidence interval in the context of the problem? a) 95% of the measurements of the equatorial radius of the planet Jupiter were between 71,462 and 71,522 km. b) We're 95% confident that the equatorial radius of the planet Jupiter is between 71,462 and 71,522 km. c) We're 95% confident that 95% of the measurements of the equatorial radius of the planet Jupiter were between 71,462 and 71,522 km. d) We're 95% confident that the mean equatorial radius of the planet Jupiter is between 71,462 and 71,522 km. e) There's a 95% chance that the mean equatorial radius of the planet Jupiter is between 71,462 and 71,522 km.

d) We're 95% confident that the mean equatorial radius of the planet Jupiter is between 71,462 and 71,522 km.

USA Today (February 17, 2011) described a survey of 1,008 randomly selected American adults. One question on the survey asked people if they had ever sent a love letter using e-mail. Suppose you want to use the results of the survey to decide if more than 20% of American adults have written a love letter using e-mail. What is the variable of interest and what type of variable is it? a) The number of American adults who have written a love letter using e-mail. It is quantitative. b) Whether or not more than 20% of American adults have written a love letter using e-mail. It is categorical c) The proportion (or percentage) of American adults who have written a love letter using e-mail. It is quantitative. d) Whether or not an American adult has written a love letter using e-mail. It is categorical. e) The proportion (or percentage) of American adults who have written a love letter using e-mail. It is categorical.

d) Whether or not an American adult has written a love letter using e-mail. It is categorical • To determine a variable of interest, think about what is recorded on one case. For each case (American adult), whether or not they have written a love letter using e-mail is what is recorded (yes/no), making this a categorical variable.

In the Lady Tasting Tea example, suppose the researchers set a significance level of 0.05 prior to performing the experiment. The lady correctly identified the preparation method of the tea in 8 out of 8 cups presented to her and the p-value for the study was 0.0039. Are the results statistically significant? a) No. Since the p-value < α , the decision is to reject the claim in the null hypothesis and say the claim in the alternative hypothesis is true, but this does not imply the results are statistically significant. b) Yes. Since the p-value < α , the decision is to NOT reject the claim in the null hypothesis and say the claim in the alternative hypothesis is true, which implies the results are statistically significant. c) No. Since the p-value < α , the decision is to NOT reject the claim in the null hypothesis and say the claim in the alternative hypothesis is true, which implies the results are NOT statistically significant. d) Yes. Since the p-value < α , the decision is to reject the claim in the null hypothesis and say the claim in the alternative hypothesis is true. This implies the results are statistically significant.

d) Yes. Since the p-value < α , the decision is to reject the claim in the null hypothesis and say the claim in the alternative hypothesis is true. This implies the results are statistically significant. • In other words, we would make the decision that the lady wasn't just guessing on each cup. Since we made the decision to reject the claim in the null hypothesis, the results are said to be statistically significant

In an alternative hypothesis, we may believe that the difference in means is a) positive b) negative c) something other than 0 d) all of the above

d) all of the above

d) convenience sample

In a hypothesis test, we calculate the probability a) that the null hypothesis is true b) of observing the evidence given that the alternative hypothesis is true c) of observing the evidence and use that to make a decision d) of observing the evidence given that the null hypothesis is true. e) that the alternative hypothesis is true

d) of observing the evidence given that the null hypothesis is true • Just like in a jury trial where the jury starts with the belief that the null hypothesis is true, a statistical hypothesis test will always start with the belief that the null hypothesis is true.

Here are the summary statistics for the original sample of 5 students: min 6 Q1 11 median17 Q3 22 max 29 mean 17 sd 9.027735 n 5 missing 0 What is the correct notation and value for the standard deviation of the 5 values in the sample? a) σx̅ = 9.0277 months b) σ = 9.0277 months c) SEx̅ = 9.0277 months d) s = 9.0277 months

d) s = 9.0277 months

A study is designed to determine whether grades in a statistics course could be improved by offering special review material. The 250 students enrolled in a large introductory statistics class are also enrolled in one of 20 lab sections. The 20 lab sections are randomly divided into 2 groups of 10 lab sections each. The students in the first set of 10 lab sections are given extra review material during the last 15 minutes of each weekly lab session. The students in the remaining 10 lab sections receive the regular lesson material, without the extra review material. The final grades of the students who reviewed weekly were higher, on average, than the students who did not review every week. 3. What is the correct conclusion for this study? a. There is no evidence to suggest that review material helps increase the grade of a statistics student taught by this instructor (p-value = 0.02) b. There is strong evidence to suggest that review material does not helps increase the grade of a statistics student taught by this instructor (p-value = 0.02) c. There is strong evidence to suggest that review material does not helps increase the grade of a statistics student taught by this instructor (p-value = 0.98) d. There is strong evidence to suggest that review material helps increase the grade of a statistics student taught by this instructor (p-value = 0.02)

d. There is strong evidence to suggest that review material helps increase the grade of a statistics student taught by this instructor (p-value = 0.02)

To account for the fact that the t-distribution will have a larger area in its tails for smaller sample sizes, a value called the _____ __ _____ will have to be calculated.

degrees of freedom • The formula for the degrees of freedom for the one-sample t-methods is n - 1

Consider the random experiment of randomly selecting one number from 1 to 10. The event A = selecting a 5 and the event B = selecting a 2. Which of the following best describes the relationship between these two events, disjoint or complementary?

disjoint - since there is only one number being selected both a 5 and a 2 cannot be selected. Therefore, the events are disjoint. They are not complementary because although they cannot occur at the same time there are more than two possible events in the sample space

What is the percentile of the upper bound of a 95% confidence interval? a) 2.5th b) 5th c) 10th d) 95th e) 97.5th

e) 97.5th

e) All American adults • The question of interest is about all American adults. The random sample is also of American adults. It makes sense that we'll be able to make a conclusion about all American adults based on the sample of 1,008 American adults.

John took a history quiz that contained 50 true/false questions. Each question had only two choices: true or false. John answered 30 questions correctly. John's teacher wondered if John's score was an indication that he knew the material or if he was just guessing. Which of the following is the correct null hypothesis? a) H0: John is just guessing. Therefore, the proportion of questions he would answer correctly is more than 0.50. b) H0: John is just guessing. Therefore, the proportion of questions he would answer correctly is 0. c) H0: John is not guessing. Therefore, the proportion of questions he would answer correctly is more than 0.50. d) H0: John is not guessing. Therefore, the proportion of questions he would answer correctly is more than 0. e) H0: John is just guessing. Therefore, the proportion of questions he would answer correctly is 0.50.

e) H0: John is just guessing. Therefore, the proportion of questions he would answer correctly is 0.50.

Recall that in Example 22.5 John got 30 of the 50 true/false questions correct and from a one-sided hypothesis test the p-value was found to be 0.0961. Which of the following is a correct interpretation of this p-value? a) The probability that the null hypothesis is true and John is just guessing on each question is 0.0961. b) The probability that the alternative hypothesis is true and John is not guessing on each question is 0.0961. c) John would answer 30 questions correctly on the 50 question exam 9.61% of the time. d) John would answer 30 questions correctly on the 50 question exam 9.61% of the time if he was just guessing on each question. e) John would answer 30 or more questions correctly on the 50 question exam 9.61% of the time if he was just guessing on each question.

e) John would answer 30 or more questions correctly on the 50 question exam 9.61% of the time if he was just guessing on each question.

A researcher collects her data and then decides to use a significance level of 0.10 on her hypothesis test. Is this legitimate to do? a) Yes b) No. She should use a much higher significance level c) No. She should use a much lower significance level d) No. She should analyze her data before deciding on a significance level e) No. A significance level must be determined prior to collecting any data

e) No. A significance level must be determined prior to collecting any data • A researcher always needs to determine a significance level prior to collecting any data! The question of interest is the first step in the Statistical Process and this is the point when a significance level should be determined if a hypothesis test is needed to answer the question of interest.

USA Today (February 17, 2011) described a survey of 1,008 randomly selected American adults. One question on the survey asked people if they had ever sent a love letter using e-mail. Suppose you want to use the results of the survey to decide if more than 20% of American adults have written a love letter using e-mail. What is the parameter in this problem and what does the parameter represent? a) μ, the proportion of American adult who have written a love letter using e-mail b) μ, the average number of American adult who have written a love letter using e-mail c) pno and pyes, where pno = the proportion of American adults who have not written a love letter using e-mail, and pyes = the proportion of American adults who have written a love letter using e-mail. d) p, the average number of American adult who have written a love letter using e-mail e) p, the proportion of American adult who have written a love letter using e-mail.

e) p, the proportion of American adult who have written a love letter using e-mail. • For categorical variables of interest, we keep track of the proportion in one of the categories. Therefore, the parameter is the population proportion. The notation for the population proportion is p

Characteristics of Confidence Intervals:

• The more certain we are that an interval captures a population parameter, the higher the level of confidence assigned to that interval • Confidence levels are generally given as percentages • Proper statistical syntax when referring to a confidence interval is to state the level of confidence first • The higher the level of confidence, the greater the number of values that need to be included in the range between the lower and upper bounds of the confidence interval • To have a confidence level of 100%, the confidence interval would need to cover all values in the sample • A confidence level of 0% implies we know for sure that the parameter is NOT one of the values listed in the confidence interval • Point estimates have very low levels of confidence because it is rare that a sample statistic (such as a sample mean) actually equals the population parameter • Typical levels of confidence are 90%, 95%, and 99%, with 95% being the most common

When interpreting any confidence interval, make sure:

• The population is included in the context of the problem • The variable is included • To include "95% confident" or "95% sure", but NEVER include "95% chance". (The word "chance" indicates a probability of the population mean being in the interval, but there cannot be a probability assigned to a fixed value! The population mean is fixed and will either be in the interval or it won't.) • To include the word "mean" or "average." A confidence interval is about a population parameter (like the mean) and not about individuals in the population. That is, a confidence interval does NOT say that we're 95% confident that a college student's body temperature is between the bounds.

Interpreting a Normal Probability Plot:

• The question we need to ask ourselves is whether it is reasonable to say that the sample could have come from a population that was normally distributed. The closer the points are to the reference line, the more comfortable we can feel saying the population data follow a normal distribution. • A reference line is drawn in diagonally on the plot. Each point represents an observation in the sample. If all the points fall on the reference line, the sample data are normally distributed. • It is rare that data will fall exactly on the reference line. • Remember, the condition that we are assessing is whether the population data follow a normal distribution.

When the population is skewed and the sample size is small:

• The shape of the resulting sampling distribution will be slightly skewed in the same direction as the population • The mean of the sampling distribution will be equal to the population mean • The standard deviation of the sampling distribution will be smaller than the population standard deviation

Comments about the distribution of sample means spread:

• The variation of the sample means is always less than the spread of the data in the population • When calculating the standard deviation of the sampling distribution, you must take the population standard deviation and then divide it by the square root of the sample size. The result will always be less than the population standard deviation itself. • The spread in the sample means decreases when larger sample sizes are taken.

The Monty Hall Problem

• Three closed doors were shown to the contestant. • The contestant knew that behind one door was a car, while behind the other two doors were goats. • The contestant would pick a door, hoping to win the car. • The host Monty Hall would reveal what was behind one of the two unpicked doors, which was always a goat. • Then he would ask the contestant whether they wanted to switch their choice to the remaining unopened door or to stick with the door he/she originally chose.

Two types of errors that can be made in a statistical hypothesis test:

• Type I Error • Type II Error

P-Value Conclusions: • p-value ≤ 0.01 • 0.01 < p-value ≤ 0.05 • 0.05 < p-value ≤ 0.10 • p-value > 0.10

• p-value ≤ 0.01 Based on the data in the study, there is strong or convincing evidence to say the claim made in the alternative hypothesis is true. • 0.01 < p-value ≤ 0.05 Based on the data in the study, there is some or suggestive evidence to say the claim made in the alternative hypothesis is true. • 0.05 < p-value ≤ 0.10 Based on the data in the study, there is little or weak evidence to say the claim made in the alternative hypothesis is true. • p-value > 0.10 Based on the data in the study, there is not sufficient evidence to say the claim made in the alternative hypothesis is true.

≥, 30 • On your number line, the observed count is greater than the hypothesized count. Therefore, we'll consider the right-side of the number line for values that are considered "as or more unusual" • Values greater than 30 are further away from the hypothesized count than 30 is from the hypothesized count. Therefore, values considered "as or more unusual" are 30 or more

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ST 351 - Statistical Methods (Combined Second Half Study Set)

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