Stats 2

two type of conclusions:

"The data provide enough evidence to reject claim 1 and accept claim 2"; or "The data do not provide enough evidence to reject claim 1."

If we are testing an alternative hypothesis of Ha: p ≠ po, which of the following test statistics will give the smallest p-value?

-2

basically 4 steps in the process of hypothesis testing:

1. State the null and alternative hypotheses. 2. Collect relevant data from a random sample and summarize them (using a test statistic). 3. Find the p-value, the probability of observing data like those observed assuming that Ho is true. 4. Based on the p-value, decide whether we have enough evidence to reject Ho (and accept Ha), and draw our conclusions in context.

Two important results that we discussed at length when we talked about the confidence interval for μ also apply here:

1. There is a trade-off between level of confidence and the width (or precision) of the confidence interval. The more precision you would like the confidence interval for p to have, the more you have to pay by having a lower level of confidence. 2. Since n appears in the denominator of the margin of error of the confidence interval for p, for a fixed level of confidence, the larger the sample, the narrower, or more precise it is. This brings us naturally to our next point.

90% level of confidence

1.645

95% level of confidence

2

99% level of confidence

2.576

A parking garage has two entrances. Let X be the number of cars that enter the garage through door A in an hour, and Y be the number of cars that enter through door B in an hour. Assuming that μx = 15 and μy = 25, what is the mean of Z, the total number of cars that enter the garage in an hour.

40

The sampling distribution of a statistic is

A distribution of a single statistic from repeated random samples of the same size, from the same population.

In which of the following situations would it not be appropriate to use a paired t-test to analyze the data?

A psychiatrist believes that a new medication (New Drug) is more effective at reducing the symptoms of depression than a placebo, sugar pill (Placebo). Sixty people with depression were randomly assigned to one of two groups: 1) New Drug or 2) Placebo and were given the assigned drug or sugar pill for 30 days. Each of the 60 participants then completed the Beck Depression Inventory and the mean depression score for the New Drug group is compared to the mean depression score for the Placebo group.

Advertising researchers claim that the power of curiosity can be harnessed to design an effective Internet advertising strategy that results in a better evaluation of the advertised product. They develop six advertising texts with varying amounts of "curiosity" triggers. College students are randomly assigned to one of the six versions of the advertisement text and their evaluation score of the advertised product is recorded. What statistical test?

ANOVA

Statistical hypothesis testing

Assessing evidence provided by the data in favor of or against some claim about the population.

interval estimation Example

Based on sample results, we are 95% confident that p, the proportion of U.S. adults who are in favor of stricter gun control, is between 0.57 and 0.63.

Point Estimation Example

Based on sample results, we estimate that p, the proportion of all U.S. adults who are in favor of stricter gun control, is 0.6.

Suppose the results indicate that the null hypothesis should be rejected; thus, it is possible that a type I error has been committed. Given the type of error made in this situation, what could researchers do to reduce the risk of this error?

Choose a .01 significance level, instead of a .05 significance level.

Central Limit Theorem applies.

First, the sample must be random. Assuming that the sample is random, recall from the Probability unit that the Central Limit Theorem works when the sample size is large (a common rule of thumb for "large" is n > 30), or, for smaller sample sizes, if it is known that the quantitative variable of interest is distributed normally in the population. The only situation when we cannot use the confidence interval, then, is when the sample size is small and the variable of interest is not known to have a normal distribution

Does second-hand smoke increase the risk of a low birthweight? A baby is considered have low birthweight if he/she weighs less than 5.5 pounds at birth. According to the National Center of Health Statistics, about 7.8% of all babies born in the U.S. are categorized as low birthweight. Suspecting that the national percentage is higher than 7.8%, researchers randomly select 1200 babies whose mothers had extensive exposure to second-hand smoke during pregnancy and find that 10.4% of the sampled babies are categorized as low birth weight. Let p be the proportion of all babies in the U.S. that are categorized as "low birth weight." Give the null and alternative hypotheses for this research question.

H 0 : p = 0.078 H a : p > 0.078

Do oddsmakers believe that teams who play at home will have home field advantage? Specifically, do oddsmakers give higher point spreads when the favored team plays home games as compared to when the favored team plays away games? Two samples were randomly selected from three complete National Football League seasons (1989, 1990, and 1991). The first sample consisted of 50 games, where the favored team played in a home game, while the second sample consisted of 50 games, where the favored team played in an away game. The oddsmakers' point spreads (which are the number of points by which the favored team is predicted to beat the weaker team) were then collected. If µ1 and µ2 represent the mean point spread for home games and away games, respectively, which of the following is the appropriate pair of hypotheses in this case?

H0: μ1 = μ2 Ha: μ1 > μ2

We would like to see if the observed linear relationship between the damage caused by a fire and the distance the fire was from a fire station can be generalized to all fires. To test this, which of the following sets of hypotheses would you use?

Ho: there is no linear relationship between distance and damage, Ha: there is a linear relationship between distance and damage

hypothesis testing

In hypothesis testing, we have some claim about the population, and we check whether or not the data obtained from the sample provide evidence against this claim.

For which of the following would it be most appropriate to use an ANOVA to analyze the data?

In order to determine the effectiveness of different diet plans, 90 people are randomly assigned to one of three groups: (1) low calorie diet, (2) low-fat diet, or (3) low-carb diet. At the end of 60 days, the number of pounds lost was recorded for each of the participants.

Point Estimation

In point estimation, we estimate an unknown parameter using a single number that is calculated from the sample data.

Suppose the results indicate that the null hypothesis should not be rejected; thus, it is possible that a type II error has been committed. Given the type of error made in this situation, what could researchers do to reduce the risk of this error?

Increase the sample size.

hypothesis testing example

It was claimed that among all U.S. adults, about half are in favor of stricter gun control and about half are against it. In a recent poll of a random sample of 1,200 U.S. adults, 60% were in favor of stricter gun control. This data, therefore, provides some evidence against the claim.

Research suggests that identical twins have many similarities, such as body weights, even as adults. A sample of ten sets of identical twins were weighed and had their weights recorded in kilograms. The researcher wishes to test whether there is a statistically significant difference in adult weight between first-born and second-born identical twins. Given the design of the study and the question of interest, which of the following 4 computer outputs is relevant to use?

Paired T-Test and CI

In a study of the impact of smoking on birth weight, researchers analyze birth weights (in grams) for babies born to 189 women who gave birth in 1989 at a hospital in Massachusetts. In the group, 74 of the women were categorized as "smokers" and 115 as "non-smokers." The difference in the two sample mean birth weights (non-smokers minus smokers) is 281.7 grams and the 95% confidence interval is (76.5, 486.9) Which gives the best interpretation of what we can conclude about the impact of smoking on birth weight?

Smoking is associated with lower birth weights. When smokers are compared to non-smokers, we are 95% confident that the mean weight of babies of non-smokers is between 76.5 grams to 486.9 grams more than the mean weight of babies of smokers.

A researcher is interested in determining whether the legibility of handwriting is related to whether the writer is left handed or right handed. A random sample of 200 students is selected from freshman English classes. Each student prepares a writing sample and it is noted whether the student is left handed or right handed. Each writing sample is judged by a panel as being "very legible", "barely readable," or "difficult to read".Which statement is true about this situation?

The null hypothesis is that there is no relationship between "handedness" and the legibility of writing.

p-value

The p-value is the probability of observing a test statistic as extreme as that observed (or even more extreme) assuming that the null hypothesis is true.

sampling distribution of the sample mean, x-bar,

The sampling distribution is normal regardless of the shape of the population distribution, as long as the sample size, n, is large enough. -The sampling distribution is normal regardless of the sample size, as long as the population distribution is normal. The sampling distribution's mean is the same as the population mean. The sampling distribution's standard deviation is smaller than the population standard deviation.

true regarding an unbiased estimator?

The sampling distribution of the unbiased estimator is centered at the parameter that it estimates.

x^2

The single number that summarizes the overall difference between observed and expected counts is the chi-square statist

A teacher is experimenting with a new computer-based instruction and conducts a study to test its effectiveness. In which situation could the teacher use a hypothesis test for matched pairs?

The teacher gives each student in the class a pretest. Then she teaches a lesson using a computer program. Afterwards, she gives each student a post-test. The teacher wants to see if the difference in scores will show an improvement.

A teacher is experimenting with a new computer-based instruction and conducts a study to test its effectiveness. In which situation could the teacher use the two-sample t-test for comparing two population means?

The teacher randomly divides the class into two groups. One of the groups receives computer-based instruction and the other group receives traditional instruction without computers. After instruction, each student takes a test and the teacher wants to compare the test scores of the two groups.

test statistic

The test statistic is a measure of how far the sample proportion pˆpˆ is from the null value p0p0, the value that the null hypothesis claims is the value of p.

a fact that would make you worried about the reliability of the results of the test in this case

Two of the six expected counts are less than 5.

Learn by Doing A double-blind experiment is conducted to investigate the side effects of hormone replacement therapy for women with menopausal symptoms. The experiment randomly assigns more than 16,000 American women to either a hormone treatment or a placebo. After five years, the HRT study finds no significant difference in the proportion of women developing breast cancer and heart disease. Researchers decide, based on this finding, to allow the study to continue.

Type 2 error

A manufacturer of rechargeable laptop batteries markets its batteries as having, on average, 500 charges. A consumer group decides to test this claim by assessing the number of times 30 of their laptop batteries can be recharged and finds the average is 497, with a standard deviation of 10. The resulting p-value is .1111; thus, the null hypothesis is not rejected. The consumer group concludes that the manufacturer's claim that its laptop batteries can be recharged, on average, 500 times is accurate. What type of error is possible in this situation?

Type II

When the variable of interest is quantitative, the population parameter that we infer about is the

When the variable of interest is quantitative, the population parameter that we infer about is the

Hypothesis testing

a four-step process in which we are assessing evidence provided by the data in favor or against some claim about the population parameter.

Chi-Square Test

c->C The chi-square test for independence examines our observed data and tells us whether we have enough evidence to conclude beyond a reasonable doubt that two categorical variables are related

Researchers question whether college students' choice of declared academic major is related to gender. which statistical test?

chi-squared test for independence

Expected Count

column total X row total / table total

Interval estimation

estimating an unknown parameter by an interval of plausible values. To each such interval we attach a level of confidence that indeed the interval captures the value of the unknown parameter and hence the name confidence intervals.

Point estimation

estimating an unknown parameter with a single value that is computed from the sample.

Two hypothesis tests were conducted. In test I, a significance level of .05 was used, and the p-value was calculated to be .025. In test II, a significance level of .01 was used, and the p-value was calculated to be .025.

in test I, the p-value is below the significance level (.025 less than .05) and therefore considered small enough to reject Ho. In test II, on the other hand, where the significance level is set at .01, the p-value is not small enough (.025 greater than .01) to reject Ho.

If column 1 is the age of the home in years, and column 2 is the price of the home (in thousands of dollars), which of the following is the appropriate inference method?

inference for regression

ANOVA F-test

is used for comparing more than two population means when the samples (drawn from each of the populations we are comparing) are independent. We encounter this situation when we want to examine the relationship between a quantitative response variable and a categorical explanatory variable that has more than two values.

paired t-test

is used to compare two population means when the two samples (drawn from the two populations) are dependent in the sense that every observation in one sample can be linked to an observation in the other sample. Such a design is called "matched pairs."

We say that a point estimator is unbiased if

its sampling distribution is centered exactly at the parameter it estimates.

margin of error is the product of two components:

m=confidence multiplier ⋅SD of the estimator

If column 1 is the price (in thousands of dollars) of a sample of five houses from ten years ago, and column 2 is the price (in thousands of dollars) of the same homes today, which of the following is the appropriate inference method?

matched pairs

To test whether Internet use increases depression score, we measure the depression scores of a random sample of non-Internet-users, have them use the Internet for a specified time, then measure their depression scores again. which statistical test?

matched pairs t-test

When the variable of interest is categorical, the population parameter that we will infer about is the

population proportion (p)

Conditions for Q-Q t test

the observed data indeed look linear (otherwise it would not make sense to try and generalize them) the observations are independent there are no extreme outliers in the data the sample size is fairly large

events A and B are independen

then P(A and B) = P(A) * P(B)

If column 1 is the price (in thousands of dollars) of a sample of five houses from ten years ago, and column 2 is the price (in thousands of dollars) of a sample of a different five houses from today, which of the following is the appropriate inference method?

two independent samples

We suspect the overall mean monthly rent of apartments in Shadyside is higher than in Oakland, so we survey a random sample of Oakland apartments, and a random sample of Shadyside apartments. What statistical test?

two-sample t-test

An NBC News/Wall Street Journal poll conducted in 2010 determined that 61 percent of Americans did not support the Tea Party movement. In a poll of 1,000 Americans this year, 64 percent say they do not support the Tea Party movement. Has opposition to the Tea Party movement increased since 2010? We tested the following hypotheses at the 5 percent level of significance: H0: The proportion of Americans this year who oppose the Tea Party movement is 0.61. Ha: The proportion of Americans this year who oppose the Tea Party movement is greater than 0.61. The P value is .026, so we reject the null hypothesis, H0 , and accept the alternative hypothesis, Ha. We conclude that public opposition to the Tea Party movement is greater than 61% this year.

type 1 error

A 2011 survey, by the Bureau of Labor Statistics, reported that 91% of Americans have paid leave. In January 2012, a random survey of 1000 workers showed that 89% had paid leave. The resulting p-value is .0271; thus, the null hypothesis is rejected. It is concluded that there has been a decrease in the proportion of people, who have paid leave from 2011 to January 2012. What type of error is possible in this situation?

type I

two-sample t-test

used for comparing two population means when the two samples (drawn from the two populations) are independent.

interval estimation

we estimate an unknown parameter using an interval of values that is likely to contain the true value of that parameter (and state how confident we are that this interval indeed captures the true value of the parameter).

confidence interval

x±z∗⋅σ/√n