STATS 1000 Exam 2 Concepts

**** examples: describing error types (see notes)

**** example: describing error types (see notes)

small p values

- are evidence AGAINST H0 - indicate the sample mean was unlikely if the hypothesized value is correct - suggests the hypothesized value may not be correct

3 things every confidence interval interpretation includes

- confidence level - description of parameter being studied - upper and lower bounds

sample space

- set of all possible outcomes of a trial - denoted by S - if there are a countable number of outcomes, use set notation (list outcomes in a set of braces) - if there are an infinite number of outcomes, use interval notation (denote upper and lower bounds inside parentheses/brackets)

What are the similarities and differences between a histogram of samples of 90% confidence intervals vs. 99% confidence intervals?

- similarities: all sample means are the same - differences: (1) 99% confidence intervals are wider than 90% confidence intervals. (2) only one 99% confidence interval missed the population mean, while ten 90% confidence intervals missed - takeaway: 99% confidence intervals are more accurate than 90% but less precise (i.e. interval gives wider range of plausible values)

general steps for calculated confidence intervals

- step 1: check for normality (at least one of the rules of thumb must apply) - step 2: calculate the interval

What is the complement of each of these events? 1. rolling a sum of less than 8 on two fair dice 2. taking at least 40 minutes to complete an exam in a MWF class 3. flipping heads more than once on two fair coins

Ac = {8,9,10,11,12} Bc = [0,40) Cc = {0,1}

law of total probability

The probability of an event is the sum of its probability across every possible condition (essentially you can add joint probabilities together)

trial

an attempt of a random phenomenon

central limit theorem

the sampling distribution of the sample mean approaches a normal distribution as the sample size increases regardless of the shape of the original population (***put the three rules of thumb on cheatsheet)

How does the standard normal table work?

the standard normal table displays the area under the standard normal curve (aka probability) to the left of the Z-score that is looked up

independent events

when the outcome of one trial does not influence or impact the outcome of another

2 rules that every probability must satisfy

(1) a probability is a number between 0 and 1 that could also be 0 or 1 (2) the probability of the set of all possible outcomes in the sample space must be 1; that is P(S) = 1

3 methods of assigning probability

(1) classical approach: assigns the same probability to each possible outcome; used often in fair games of change (2) empirical approach: long-run frequency with which an outcome occurs; take total number of observations for an outcome and divide by total number of observations (3) subjective approach: degree of belief that we have in the occurrence of an event; used when classical approach is unreasonable and no information exists to calculate proportions

How can a confidence interval for the population mean become narrower?

(1) decrease confidence level: however this comes with the drawback of lower accuracy of the interval (2) decrease standard deviation: however this is not always possible (3) increase sample size: usually the best option

2 requirements of sample spaces

(1) exhaustive: includes all possible outcomes (2) disjoint: two outcomes cannot occur simultaneously on the same trial

how to calculate normal distribution percentiles

(1) figure out: what Z- score corresponds to the desired area in the tail? to solve, find the percentile inside the standard normal table and solve for the Z-score by moving left and up (2) given the Z-score from step 1, what value is this many standard deviations above/below the mean? to solve, use the mean, standard deviation, and Z-score from step 1 to unstandardized Z = (x - mu)/sigma and solve for X

**** example: choice of hypothesized mean (see notes)

**** example: choice of hypothesized mean (see notes)

**** example: choice of population (see notes)

**** example: choice of population (see notes)

**** example: choice of sample size (see notes)

**** example: choice of sample size (see notes)

**** example: ramifications of errors (see notes)

**** example: ramifications of errors (see notes)

**** examples of one-sided vs. two-sided tests (see notes)

**** examples of one-sided vs. two-sided tests (see notes)

**** examples: appropriate level of significance (see notes)

**** examples: appropriate level of significance (see notes)

**** examples: identifying error types (see notes)

**** examples: identifying error types (see notes)

*****example p-value (see notes)

*****example p-value (see notes)

*****example test statistic (see notes)

*****example test statistic (see notes)

*****example: interpreting the results (see notes)

*****example: interpreting the results (see notes)

****Danger of Running Multiple tests (see notes)

****Danger of Running Multiple tests (see notes)

***Example: lower one sided test (see notes)

***Example: lower one sided test (see notes)

***Example: two sided test (see notes)

***Example: two sided test (see notes)

other equations for mean and standard deviation

*if X is a binomial random variable* with "n" trials and probability of success "p", then: mean = mu = np --> because the average (expected) number of successes is the number of trials times probability of a success on any given trial) s.d. = sigma = square root of np(1-p) --> represents how much we would expect the actual number of successes to deviate from mu by

Addition rule for disjoint events

*if* A and B are disjoint, then P(A or B) = P(A) + P(B)

example of creating a sampling distribution: scenario: take a random sample of SAT scores of a given sample size, calculate the sample mean, and plot this new sample mean in a histogram. repeat 1000 times. SAT scores follow a normal distribution, with a population mean mu = 1000 and population standard deviation sigma = 217. Create three different sampling distribution histograms. First time, take samples of size 1. Second time, take samples of size 9. Third time, take samples of size 100. What changes/remains the same?

- *mean of the* sample means (aka the mean of the sampling distribution) were all nearly equal to the population mean (1000) - as you increase sample size, you decrease the spread of sample means - as you increase sample size, you decrease the standard deviation of x bar from mu - shape of sample means were all approximately normal (unimodal and symmetric) - takeaway: when sampling from a normal population, the mean of the sampling distribution equals the population mean and the shape will be normal, but the spread will shrink as n increases

mean of a probability model

- AKA expected value - denoted as mu, because we know the exact probabilities of each outcome so we can calculate the population mean - the average outcome from a probability model if the random experiment is repeated ad infinitum - mu = x1p1 + x2p2 + ... + xkpk

alternative hypothesis

- HA - the conclusion we come to if the collected evidence indicated the null hypothesis may be false - three different types: upper one-sided, lower one-sided, two-sided alternatives

normal probability plot

- a plot of the observations' Z-scores in increasing order (y-axis) against the Z-score the observation would have if the data were perfectly normally distributed (X- axis) - concave down = left skewed data - close to straight line = normally distributed data - concave up = right skewed data

standard normal random variable

- a random variable with a standard normal distribution, denoted by the symbol Z - can be created with a formula: Z = (x - mu)/ sigma = (observation - mean)/ standard deviation - this creates a Z-score which can be used to calculate probabilities using standard normal table

standard normal distribution

- a special case of the normal distribution that has a mean of zero and a standard deviation of 1 - denoted by Z - values referred to as Z -scores or Z- statistics - probabilities found using a standard normal table

random variable

- a variable whose possible outcomes are numeric values and whose probabilities are assigned according to some function or rule - generally denoted by capital letters - X, Y, and Z are most common names - Z reserved for standard normal random variables - two types: discrete and continuous

null hypothesis

- an initial guess made about a parameter (H0, H-naught) - e.g. H0: mu = mu0, where mu is parameter and mu0 is the hypothesized value of the unknown parameter - note: mu0 will be replaced with the hypothesized value in both the null and alternative hypotheses

Confidence interval

- an interval of plausible values for an unknown parameter that is calculated from the responses in a sample - provides us with a range of values that could be the true parameter - usually looks like this: statistic +/- margin of error (margin of error = critical value time standard error)

Example: measuring accuracy Scenario: IQ scores are normal with mu = 100 and sigma = 15. Take a random sample of size 30 and calculate a 95% confidence interval. Repeat this process 19 more times. Question: How many of these 20 confidence intervals would we expect to contain 100?

- answer: 19 - confidence intervals are NOT perfect - confidence levels can be interpreted as probabilities - 95% confidence literally means 95% of the time, the interval will contain the true population, but 5% of the time it will miss

large p values

- are evidence SUPPORTING H0 - indicate the sample mean is not terribly unusual if the hypothesized value is correct - suggests that the hypothesized value could be plausible as the true parameter

Scenario: SAT scores are normally distributed with population mean mu = 1000 and population standard deviation sigma = 217. A random sample of 25 Pitt students finds a sample mean SAT score of 1250. Question: What three things do you need to continue with a hypothesis test with this scenario?

- check if the shape of the sampling distribution is normal (necessary to continue with the hypothesis test) - calculate the test statistic ("how many standard errors is 1250 from the hypothesized mean of 1000?") - calculate the p value ("how unusual is this sample mean of 1250 if the true population mean SAT score of Pitt students is actually 1000?")

relationship between hypothesis tests and critical values

- confidence interval (CI): provides a range of plausible values for an unknown parameter - hypothesis test: determines if a hypothesized value is plausible - CI result 1: CI contained hypothesized value --> hypothesized value is a plausible estimate of the parameter (fail to reject null) - CI result 2: CI lies entirely above hypothesized value --> true parameter is significantly greater than the hypothesized value (reject null) - CI result 3: CI lies entirely below hypothesized value --> true parameter is significantly less than the hypothesized value (reject null) - this is usually true but exceptions occur (super rare tho)

normal distribution

- continuous probability distribution that describes data whose histogram is unimodal and symmetric - defined by two parameters, mean (mu) and standard deviation (sigma) - entire area under curve equals 1 --> areas are interpreted as probabilities

standard deviation of a probability model

- denoted as sigma because we know the exact probabilities of each outcome so we can calculate the population standard deviation - the measure of the typical distance an outcome is away from its expected value - sigma = the square root of the sums of p1 (x1-mu)^2 + p2 (x2-mu)^2 + .... + pk(xk-mu)^2

binomial distribution

- discrete probability distribution that calculates the probability of observing a certain number of successes in a fixed number of independent trials - n = trials, k = successes, p = success probability - note: 0! = 1

disjoint events

- events that cannot both occur at the same time - for two events A and B, P (A and B) = 0 - A occurring prevents B from occurring and B occurring prevents A from occurring

how do we find the z-scores that bound the middle 95% of the standard normal distribution?

- find the 2.5th and 97.5th percentiles (note: we can use the standard normal distribution because we know that the shape of the sampling distribution of SAT score is normal AKA passes at least one of the rules of thumbs for the central limit theorem) - remember that every confidence interval has two critical values. because we are finding the z-scores that bound the middle 95%, we need Z 0.025 (which is -1.96) and Z 0.975 (which is +1.96)

How to calculate a 95% confidence interval

- find the values you need: (1) statistic, (2) standard error, which remember is sigma/square root of n b/c we are dealing with a sample, and (3) critical value, which is equal to the Z-scores bounding the middle 95% of the standard normal distribution

Scenario: You have a 23% chance of winning some amount of money on a particular scratch-off the lottery ticket. Suppose you buy 6 of these tickets and want the probability of 1 or 2 winners. Why is this a binomial distribution problem?

- fixed number of trials (6) - 2 possible outcomes (winner or loser) - fixed probability of success (0.23) - tickets are independent - want: probability of a given # of successes

sampling distribution

- for quantitative data, this is the distribution of all possible sample means for a given mean (mu), standard deviation (sigma), and sample size (n) - defined by three components: (1) mean, (2) standard error, which is the standard deviation of the sampling distribution, which measures how spread out the sample *means* tend to be, and (3) shape - in other words, this describes where we can expect a sample mean to fall in relation to the population mean

test statistic for testing a population mean

- for testing a population mean, the test statistic is Z = (x bar - mu0) OVER (sigma / square root of n) - Z = the number of standard errors that separate the sample mean and hypothesized mean

probability tree

- graphical technique that displays the relationship between two dependent variables - first "split" comes from events that do not depend on anything - second "split" dependent upon result of first event - joint probabilities calculated by multiplying along each path

general multiplication rule

- if A and B are any two events, then P(A and B) = P(A) * P(B|A) - note: you can only multiply the individual probabilities if the events are independent, so if one event impacts the probability that another event occurs, you must use this rule - the general multiplication rule is just the conditional probability equation rearranged to equal P (A and B)

how p value affects our decisions

- if the p value is small, then we REJECT the null hypothesis and side with the alternative hypothesis (i.e. the hypothesized value is not plausible as the true parameter) - if the p value is large, then we FAIL TO REJECT the null hypothesis and conclude that the hypothesized value is plausible as the true parameter - *after* specifying a level of significance (alpha), determine if the p value is less or greater than alpha. If p value is less than alpha, reject null (the result is statistically significant). If p value is greater than alpha, then fail to reject null

sample size calculations

- in many fields, researchers know ahead of time the acceptable margin of error (i.e. the precision) and the confidence level to be used (i.e. accuarcy) - only way to change the margin of error is through the sample size - sample size: given a level of confidence 100(1-alpha)% and a population standard deviation sigma, the necessary sample size to attain a desired margin of error "m" is: n = the value (Z sub alpha over two times sigma / m) squared

Law of Large Numbers

- it becomes increasingly difficult for x bar to deviate from mu as the sample size increases - to see this law in action: sample n observations independently from any population with mean mu. calculate the sample mean x bar - as n increases, the sample mean x bar converges to the population mean mu - occurs because in small samples, one outlier drastically skews the mean up or down. but in large samples, an unusually small observation will often be balanced by an unusually large observation

Scenario: An urn contains 12 black balls and 8 white balls. Select one ball, look at the color, do not replace it, and select another ball. Question: What is the probability that both balls selected are black?

- let F = 1st ball black and S = 2nd ball black - P(F) = 12/20 = 0.60 - F and S are not independent because the first ball is not replaced - P(S|F) = 11/19 = 0.579 (b/c 11 black balls left out of 19) - P (F and S) = P(F) * P(S|F) = (0.60)(0.579) = 0.3474

example: looking ahead to making inference Scenario: suppose we want to learn about the average SAT score for only incoming freshman at Pitt. This will likely be higher than the national mean of 1000. Question: how can we determine if Pitt students score higher than 1000 on average if we don't know their population mean?

- make a guess about the true population mean - take a random sample of incoming Pitt freshman - create a histogram and find summary statistics (mean, standard deviation) to determine if the shape of the sample mean is normal - compare the sample mean with our guess for the population mean to see if our guess is plausible

mean and standard error of sample mean

- mean: mu of x bar = mu - standard error: sigma of x bar = sigma / square root of n - if central limit theorem holds (i.e. the shape of the sampling distribution is normal), then the sample mean is standardized as Z = (X bar - mu) / (sigma / square root of n)

confidence level

- measure of how certain we are that the confidence interval contains the true population parameter - denoted by 100(1-alpha)% whre alpha is the total area being left out

Scenario: SAT scores are normally distributed with population mean mu = 1000 and population standard deviation sigma = 217. A random sample of 25 Pitt students finds a sample mean SAT score of 1250. Question: What is the sampling distribution of the sample mean?

- muX = 1000 (in inference, use the hypothesized mean, because we need to know how unusual the sample mean is relative to what we believe the mean could be) - standard error sigmaX = 217/square root of 25 = 43.4 - shape = normal (b/c of rule of thumb #1)

critical value

- multiplier in a confidence interval that tells us how many standard errors to extend in each direction from the statistic to find the upper and lower bounds of the confidence interval - note: critical values change depending on (1) the confidence level and (2) the variable situation

intersection

- occurs when *both* event A and event B occur - probabilities usually get multiplied --> smaller value because it is harder for two events to both occur - probability of an intersection is called a joint probability

union

- occurs when *either* event A or event B occurs - "OR" - probabilities get added --> larger value because we only need at least one of the two events to occur - includes situations where both events occur

type II error

- occurs when the null hypothesis is false in reality, but the evidence tells us to fail to reject H0 - common reason: hypothesized value was wrong, but the sample did not pick up enough evidence to reject that value

type I error

- occurs when the null hypothesis is true in reality, but the evidence tells us to reject H0 - common reason: a bad sample resulted in a small p-value

test statistic

- one criterion used to decide if a hypothesized value is a plausible value for the parameter - measure of how different the statistic is from the hypothesized value of the parameter - calculation changes depending on the type of test - the more extreme this is, the smaller the p value

summary one-sample Z-test for a population mean

- one-sample Z tests are used to perform inference on a single unknown population mean when the population standard deviation (sigma) is known - conditions: shape of sampling distribution of sample mean must be normal - test statistic: Z = (x bar - mu0) OVER (sigma / square root of n) - confidence interval: X bar plus or minus Z (a different one; this is the critical value that matches the level of significance) times (sigma/square root of n)

types of estimates

- point estimate: a single value that is provided as the estimate of an unknown parameter in a population (x bar = point estimate for mu, s = point estimate for sigma, p hat = point estimate for p) - interval estimate: an interval of plausible values for an unknown parameter. Based on the sample, each value could reasonably be the value of the parameter

Scenario: Suppose 25% of all hikers on a trail see a bear. Of those who see a bear, 70% post a review of the trail online while only 20% of those who do not see a bear post a review. Question: What information do we already know?

- posting a review depends on seeing a bear (so first split will be on seeing a bear. second split will be on posting a review) - probability of seeing a bear = P(B) = 0.25 - probability of posting a review given seeing a bear = P (R|B) = 0.70 - probability of posting a review given NOT seeing a bear = P(R|Bc) = 0.20

conditional probability

- probability that one event (B) will occur given that we already know the outcome of another possibly related event (A) - restricts the sample space to only those outcomes that we know have initially occurred - P(B|A) = P (A and B) / P(A) --> the probability of B *given that A has happened/is happening

Cautions about hypothesis testing

- researchers have a decent amount of control over how significant their results are as they can often choose: the population being studied, the form of alternative hypothesis, value of the hypothesized mean, and sample size - it is also possible to run multiple analysis on different samples of the data or run multiple analysis using different hypotheses, but only report the desired results (this is called p-hacking and is extremely unethical)

example of creating a sampling distribution, involving the central limit theorem: scenario: a fair six-sided die has a population mean of mu = 3.5 and a standard deviation of sigma = 1.708. Roll fair six-sided die, calculate and plot the sample mean. Repeat this 1000 times to create one histogram. Create three histograms for sampling distributions of rolling (1) one die, (2) four die, (3) sixteen die. What changes/remains the same?

- sampling distribution means were all nearly equal to the population mean of 3.50 - as the sample size increased, standard error become smaller - as the sample size increased, the shape become more normal in time - takeaway: when sampling form a slightly skewed or uniform population, the mean of the sampling distribution always equals the population mean, but the shape will become normal and the spread will shrink as n increases

Scenario: SAT scores are normally distributed with population mean mu = 1000 and population standard deviation sigma = 217. Question: how can we determine if 1000 is a plausible value for the average SAT score of Pitt students?

- solution 1: take a random sample of SAT scores from Pitt students, find the sample mean, and see if it equals 1000 - problem 1 with solution 1: cannot sample the entire population so we will not find mu (average of all Pitt students scores) - problem 2: variability exists from sample to sample so the sample mean x bar will not be a perfect representation of the population mean mu - problem 3: cannot display the effect of taking larger spaces - solution 2 is better: find an interval of plausible values for what the mean SAT score of Pitt students could be - solution 3 could also work: determine how unusual the sample mean would be under the assumption that the true population mean SAT score of Pitt students is 1000. Then perform a hypothesis test

probability distribution

- the collection of all possible outcomes of a random variable and their corresponding probabilities - can be organized in a table or calculated through a formula/function - same rules for generic sample spaces must hold (each probability must be between 0 and 1, sum of all probabilities must sum to 1)

level of significance

- the cutoff that signifies that a p-value is small enough to reject the null hypothesis - denoted by alpha and usually set to either 0.01, 0.05, or 0.10

error types and level of significance

- the level of significance and the errors types are related - 5% is a commonly used level of significance that will suffice in a majority of situations. However, your choice of a level of significance may also be dictated by if a type 1 or type 2 error is worse - if a type I error is worse, lower the level of significance - if a type 2 error is worse, raise the level of signficiance - if neither error type is clearly worse, stick iwth 5%

p-value

- the probability of obtaining data as extreme or more extreme than what was originally observed assuming the null hypothesis is true - calculated in different ways depending on the alternative hypothesis

marginal probability

- the probability that an individual event occurs regardless of the outcome from any other variable - values contained in the margins of the table where the entries in the desired row/column have been (row sums/column sums)

joint probability

- the probability that the intersection of two events occurs - values contained inside the table in the intersection of the row and column of desired events

3 types of alternative hypotheses

- upper one-sided alternative = HA: mu > mu0 = parameter is greater than hypothesized parameter - lower one-sided alternative + HA: mu < mu0 = parameter is lower than hypothesized parameter - two-sided alternative = HA: mu does NOT equal mu0

how to calculate p value based on the type of alternative hypotheses

- upper one-sided: area in upper tail, above test statistic - lower one sided: area in lower tail below test statistic - two-sided: total area in both tails beyond positive and negative values of test statistic

Importance of results: Why does the p-value not indicate whether a result is important?

- we can force a small p-value by simply taking a large enough sample - if we shrink the standard error (sigma/square root of n) by increasing n, then the test statistic Z = (x bar - mu0) OVER (sigma / square root of n), will increase - small p-values result from larger test statistics - e.g. a difference of .06 mph in baseball is not really noteworthy

benefits and drawbacks of smaller p-values

- we normally like to have narrower confidence intervals as they provide a more precise estimate of the parameter - but we do not necessarily always want to attain a smaller p -value - e.g. for the Pitt SAT scores, we wanted a small p-value to show that Pitt students scored better than the national average - e.g. for male height, p- value really didn't matter because nothing important was coming from the result as we were just seeing if the average height was less than 72 inches - e.g. for screw length, we wanted a large p-value to show that the machine was under control and making screws the correct length

Example: impossible probabilities

- while you can know the mean and standard deviation, if you do not know the shape (distribution) of the sampling distribution, you cannot compute the probability of the average of a specific sample size being less than or greater than a specific value - note: to be able to compute the probability you would just need to be able to apply one of the rules of thumb (e.g. might just need to increase sample size to at least 30, even if data skewed)

Assessing Normality

-in probabillity, we often know if a variable is normally distributed based on having a large amount of historic data - allows us to calculate probabilities and percentiles immediately - in inference, we only work with samples. Before analyzing the data, it is often important to determine if a variable is normal by analuzing the sample - the normality of a sample can be assessed using a normal probability plot; histograms also work but may not paint a clear picture for small samples

Scenario: A hotel asks its customers to rate the cleanliness of its rooms on a 5-point scale. The probabilities are the following: poor = 0.08. fair = 0.12. average = 0.18. good = 0.25. excellent = 0.37

0.92 (because '"at least fair" = everything except what is below fair)

Scenario: A 2021 Gallup poll asked respondents for their political affiliation and whether they were concerned about global warming. (Results are in a table in notes so can't give formula vs. numerical answer). 1. Which variable depends on the other? 2. What will the probability tree look like? 3. What probabilities need to be represented in the tree?

1. Global warming opinion depends on political party 2. three branches at the first split for political affiliation, two branches from each political affiliations for concern level. Total is 6 combinations 3. P (D), P (R), P(I), P(C | D), P (C | R), P (C | I). note: you find P(Cc | D/R/I) based on P (C | D/R/I)

Scenario: A 2021 Gallup poll asked respondents for their political affiliation and whether they were concerned about global warming. (Results are in a table in notes so can't give formula vs. numerical answer). 1. What is the overall probability that a person is concerned about global warming? 2. Given that a person is concerned about global warming, what is the probability that they are an Independent?

1. P(C) = P (C and D) + P (C and R) + P (C and I) 2. P (I | C) = P (C and I) / P (C)

In a deck of cards, there are 13 cards in each of 4 suits. You draw one card from the deck: 1. What is the probability of picking a king? 2. What is the probability of picking a club? 3. Is the probability of drawing either a king or a club 17/52?

1. P(K) = 4/52 2. P(C) = 13/52 3. no because events are NOT disjoint

Example: sample space Question: what is the sample space in each of the experiments? 1. roll two dice and add the results 2. measure the amount of time in minutes it takes a student to complete an exam during a MWF class at Pitt 3. flip two coins and look at the order of the results 4. flip two coins and count how many heads occurred

1. S= {2,3,4,..., 12} 2. *[*0,50*]* 3. {HH, HT, HH, TT} 4. {0,1,2}

hypothesis testing procedure

1. determine the appropriate test to use 2. determine the null and alternative hypotheses 3. collect the data 4. ensure the conditions for performing the test hold 5. calculate the test statistic 6. calculate the p value 7. calculate a confidence interval that matches the level of significance 8. write a conclusion using the above results

Scenario: Suppose 25% of all hikers on a trail see a bear. Of those who see a bear, 70% post a review of the trail online while only 20% of those who do not see a bear post a review. 1. How should the events "Seeing a bear" and "posting an online review" be described? 2. Which event depends on the other, if at all? 3. What visual representation can better display these relationships?

1. neither disjoint nor independent 2. posting a review depends on seeing a bear 3. probability tree

Scenario: SAT scores are normally distributed with population mean mu = 1000 and population standard deviation sigma = 217. 1. Ask 1 random college freshman what their SAT score was. Should we be surprised if their score was greater than 1100? 2. Ask 9 random college freshmen what their SAT score was. Should we be surprised if the sample mean SAT score was greater than 1100? 3. How must we approach finding the exact probability for the occurrence in question 2? 4. Ask 100 random college freshmen what their SAT score was. Should we be surprised if the sample mean SAT score was greater than 1100?

1. no 2. somewhat 3. we must keep in mind that we are working with the *average* of a sample 4. yes (in this example, we would have expected ~50 to score above 1000 and ~50 to score below in a large sample)

Example: interpreting confidence interval: Scenario: A 95% confidence interval for the mean SAT score of Pitt students from our sample of 25 is (1164.94, 1335.06). 1. What does this confidence interval mean in context? 2. What does the margin of error of 85.06 mean?

1. we are 95% confident that the true population mean SAT score of Pitt students is between 1164.94 and 1335.06 2. while 1250 is the best estimate, we wouldn't expect the true mean SAT score of Pitt students to deviate from this value by more than 85.06 points

Example: point estimate: Scenario: take a random sample of 25 Pitt students and find a sample mean SAT score of 1250. 1. What notation and value should be used to represent a point estimate of the average SAT score of Pitt students? 2. What notation and value should be used to represent the true population mean SAT score of Pitt students?

1. x bar = 1250 2. mu = unknown value

scenario: there are 50 marbles in a bag of four diff colors: blue, red, green, and yellow. Select one marble from the bag question 1: What is the sample space? question 2: what must be true about the blue, red, green, and yellow marbles?

1. {B, R, G, Y} 2. each color must have a probability of being selected between 0 and 1. And P(B) + P(R) + P(G) + P(Y) = 1

In a deck of cards, there are 13 cards in each of 4 suits. You draw one card from the deck: What is the probability of drawing either a king or a club?

16/52

Example: null and alternative hypotheses Scenario: SAT scores are normally distributed with population mean mu = 1000 and population standard deviation sigma = 217. We want to determine if the average SAT score of Pitt students is higher than the national average. Question: what are the null and alternative hypotheses?

H0: mu = 1000 HA: mu > 1000

which of these events are disjoint? I. rolling an odd number and an even number II. rolling an odd number and rolling a number less than 4 III. rolling a number greater than 4 and a number less than 4

I and III

Which of these pairs of events are independent? I. selecting a card from a deck, not replacing it, and drawing a 2nd card II. flipping a coin two times in a row III. the grade you receive in calc 1 and calc 2

II only; because (1) not replacing the card makes all the other cards more likely to be chosen the 2nd time, (2) coins have no memory. The first coin flip has no impact on the second, (3) your grade in calc 1 is likely a very good predictor of your calc 2 grade

Scenario: IQ scores have a normal distribution with mean 100 and standard deviation 15. What proportion of people have an IQ score between 82 and 124? (actually calculate)

P (82 < X < 124) = P ( -1.20 < Z < 1.60) = P (Z < 1.60) - P(Z < -1.20) (find area to the left of highest value. subtract area left of lowest value from that AKA subtract the area form the lowest value to -infinity)

General Addition rule

P (A or B) = P(A) + P(B) - P(A and B)

Bayes' Rule

P (B | R) = P (B and R) / P (R)

Scenario: Suppose 25% of all hikers on a trail see a bear. Of those who see a bear, 70% post a review of the trail online while only 20% of those who do not see a bear post a review. Question: Given that a person posted a review, what is the probability they saw a bear?

P (B|R) = P (B and R) / P(R) = 0.175/0.325 = 0.5385

Scenario: there are 50 marbles in a bag: 23 blue, 15 red, 8 yellow, and 4 green. What is the probability of picking a red or blue marble?

P (R or B) = P(R) + P(B) = (15/50) +(23/50) = (38/50) = 0.76

Scenario: 4% of all emails that are received are spam. Suppose you receive 20 emails in one day. What is the probability that at least 1 message is spam? (actually calculate)

P (X ≥ 1) = 1 - P (X = 0) b/c no spam messages is the compliment of at least 1 = 0.5580

Flip four coins and record each result. 16 possible combinations, each with probability of 1/16 of occurring. What is the probability of flipping heads at least once?

P (at least one head) = 1 - P( zero heads) = 1-(1/16) = 15/16

Scenario: You have a 23% chance of winning some amount of money on a particular scratch-off the lottery ticket. Suppose you buy 6 of these tickets. What is the probability that 1 or 2 tickets are winners? (actually calculate)

P (x = 1 or x = 2) = 0.6524

conditional probability rule

P(B | A) = P(A and B) / P(A) (note: must be used when one event impacts the probability that another event occurs. you must use this instead of just multiplying the two individual probabilities)

Scenario: in one lottery game, three digits form 0-9 are selected at random and combined to make a three-digit number. Players win some amount of money if they match at least one digit. How can a person win?

P(W) = 1- P(loss) = 1 - P (X and X and X) = 1 - (.90)(.90)(.90) = 0.271

Scenario: 10% of the world's population is left handed. Randomly sample 3 people and record their handedness. Question: What is the probability that you select exactly one left-handed person? (actually calculate the number)

Problem: several ways of selecting exactly one left-handed person. Cannot just calculated (.10)(.90)(.90) using the multiplication rule for independent events. (note: the picks are independent of one another because billions of ppl in the world) Solution: P(x = 1) = 0.243. (identify all possible combinations and add probabilities together and create a weird probability tree. Add another set of branches for each additional trial. Identify all combinations that satisfy the event. Sum the probabilities)

Scenario: Suppose 25% of all hikers on a trail see a bear. Of those who see a bear, 70% post a review of the trail online while only 20% of those who do not see a bear post a review. Question: What is the probability that a person posted a review?

Problem: the only probabilities that involve posting a review are condition probabilities. Answer: break down posting a review into two parts. P(R) = P(B and R) + P(Bc and R) because every person who posted a review either saw a bear or did not P (R) = 0.175 + 0.15 = 0.325

how can we get a smaller p- value in a hypothesis test?

Remember the test statistic Z = (x bar - mu0) OVER (sigma / square root of n). We need the test statistic to be larger in magnitude so we could do this by: - increasing difference between sample mean and hypothesized mean (this is more easily attained by choosing a different hypothesized value) - decrease standard deviation (not always possible) - increase sample size (usually the best option but must be done carefully)

The amount of time college students study outside of class per week is normal with a mean of 24 hours and standard deviation 8 hours. How could a normal distribution be used to find the probability of a student studying less than a specific number of hours per week?

We need to standardize because there is no easy-to-use function or rule that calculates normal distribution probabilities. We need to standardize every normal random variable to find how many standard deviations the observation is from the mean. Use the standard normal table to find probabilities.

Interpretation of sample size calcualtion

_________ must be sampled and measured to estimate the true population of ___________ to within __________.

probability table

a cross classification table between categorical variables where the entries represent the probability of the intersection of each combination of events; sum of all probabilities equals 1

Why would we want a narrower interval?

a narrower interval provides a more precise estimate of the parameter; we have a better idea of what the parameter's true value is

discrete random variable

a random variable that can take on a countable # of outcomes

continuous random variable

a random variable that can take on any numeric value

hypothesis test

an inferential procedure that tests whether a hypothesized value for a parameter is plausible

event

any combination of outcomes, typically denoted by a capital letter or word (ex: rolling an even number 2,4, or 6; selecting any spade)

random phenomenon

any process that leads to one of several potential results where the result cannot be predicted, but whose results have a regular distribution after many repetitions (ex: rolling a die, flipping a coin, selecting a card from a full deck)

In the United States, 44% have type O blood, 42% of people have Type A blood, 11% have type B blood, and 3% have type AB blood. Suppose two people are selected randomly. What is the probability both people have Type A blood?

blood types are independent so we can use the multiplication rule; P (A and A) = P(A) x P(A) = (.42)(.42) = 0.1764

Measure the number of miles a set of new tires last on a car before needing to be replaced. What type of random variable is X?

continuous; because there are an infinite number of outcomes in sample space [0, ∞)

Flip a coin 5 times. Let X be the random variable denoting the number of heads. What type of random variable is X?

discrete; because there are a countable number of outcomes in sample space {0,1,2,3,4,5}

how to convert an observation between two distributions

find the Z-score for that one observation in the distribution it is from. unstandardize using the distribution of the other (new) distribution. (answer = mean of new distribution + the product of that Z-score and the s.d. of new distribution)

multiplication rule for independent events

if A and B are independent, then P (A and B) = P(A) * P(B)

outcome

potential result of a random phenomenon (ex: rolling a 4, flipping heads, selecting the 7 of spades)

Scenario: 10% of the world's population is left handed. Randomly sample 3 people and record their handedness. Question: What is the probability that you select exactly four left-handed people?

problem: too many combinations to make a probability tree and too-time consuming to identify the ones with four successes (probability tree with 7 trials, yields 128 combinations of successes and failures) solution: use the binomial distributions

complement

set of all outcomes not included in an original event A, and denoted by A^c

Importance of results: why might a study with a non-significant result still be important?

studies with small sample sizes have difficulty achieving significance, but are often used as pilot studies to show proof of concept

rationale for the general addition rule

subtracting off the probability of the intersection accounts for the fact that it was originally included both times when we added the individual probabilities together (remember yo u MUST use this rule if events are NOT disjoint)

probability

the chance that an event occurs

complement rule

the probability that an event occurs is 1 minus the probability that it does not occur; that is P(A) = 1 - P(A^c)

Example: Inadequacy of point estimate: Scenario: take a random sample of 25 Pitt students and find a sample mean SAT score of 1250. Question: Why is a point estimate inadequate for definitively concluding that the true mean SAT score of all Pitt students is greater than 1000?

the variability in the population is not considered (a diff. sample would yield a diff. sample mean, possibly one closer to 1000. Also 25 is a relatively small sample so our sample means might be spread out enough to conclude that 1000 is still a plausible mean)

In the United States, 44% have type O blood, 42% of people have Type A blood, 11% have type B blood, and 3% have type AB blood. Suppose two people are selected randomly. What is the probability both people have the same blood type?

there are 4 different ways this event can be satisfied (important to keep in mind that blood types are independent so you should multiplication rule to find each individual intersection, e.g. P (O and O)); P (same) = P(O and O) + (A and A) + P (B and B) + P (AB and AB)

confidence interval for a single population mean

to estimate a single unknown population mean mu using a confidence interval, calculate: x bar +/- Z sub alpha/2 (sigma/ square root of n)

example: union and intersection scenario: flip two coins and record if they land on heads or tails question: which of the following events are unions? Which are intersections? I. Both flips are heads II. at least one flip is heads III. neither flip is heads

union: II only intersection: I and III

How would you solve the question: "what is the probability the high temperature on one June 7 is at least 10 degrees away from the mean?"

use Z table to find area under curve of lowest tail and multiply by 2 (because area under curve of upper tail should equal that of lower tail)

Comparing one-sided and two-sided tests

when deciding on the form of the alternative, consider the following: - the form of the alternative does NOT impact the test statistic (test statistic measures how far the sample mean is from the hypothesized mean) - form of the alternative impacts the p-value (two-sided alternative will always be twice the value of corresponding one-sided tests) - alternative hypothesis should be chosen based on what the researcher suspects is true before collecting the data (cannot collect the data/find the sample mean and then proceed to choose the alternative) - if a one-sided test is not the distinctly obvious choice, hedge with a two-sided test (two-sided test is more conservative but does not bring the risk of choosing the "wrong" direction)