Deck 12: Inference About Comparing Two Populations, Part 1

Question 1

A calculus professor wanted to test whether the grades on calculus test were the same for upper and lower classmen. The professor took a random sample of size 12 from each group. For this situation, the professor should use a matched pairs t-test.

Answer 1

False

Question 2

The sampling distribution of is normal if the sampled populations are normal, and approximately normal if the populations are nonnormal and the sample sizes n₁ and n₂ are large.

Answer 2

True

Question 3

When the population variances are unequal, we estimate each population variance with its sample variance. Hence, the unequal-variances test statistic of is approximately Student t-distributed with n₁ + n₂ - 2 degrees of freedom.

Answer 3

True

Question 4

In comparing the difference in means with a matched pairs experiment, the variable under consideration is , where the subscript D refers to the difference.

Question 5

The pooled-variances t-test requires that the two population variances need not be the same.

Question 6

The expected value of is .

Question 7

Two measurements from the same individuals is an example of data collected from a matched pairs experiment.

Question 8

In testing the difference between two population means using two independent samples, we use the pooled variance in estimating the standard error of the sampling distribution of the sample mean difference if the populations are normal with equal variances.

Question 9

The variance of is .

Question 10

The best estimator of the difference between two population means is the difference between two sample means .

Question 11

In testing the difference between two population means using two independent samples, the sampling distribution of the sample mean difference is normal if the sample sizes are both greater than 30.

Question 12

The equal-variances test statistic of is Student t-distributed with n₁ + n₂ degrees of freedom, provided that the two populations are normal.

Question 13

Independent samples are those for which the selection process for one is not related to the selection process for the other.

Question 14

Two samples of sizes 25 and 20 are independently drawn from two normal populations, where the unknown population variances are assumed to be equal. The number of degrees of freedom of the equal-variances t-test statistic is 44.

Question 15

A matched pairs experiment decreases variability (compared to two independent samples).

Question 16

Both the equal-variances and unequal variances test statistic and confidence interval estimator of require that the two populations be normally distributed.

Question 17

When we test for differences between the means of two independent populations, we can only use a two-tailed test.

Question 18

When the sample sizes are equal, the pooled variance of the two samples is the average of the two sample variances.

Question 19

Unless we can conclude that the population variances are equal, we cannot use the pooled variance estimate.

Question 20

A political analyst in Iowa surveys a random sample of registered Republicans and compares the results with those obtained from a random sample of registered Democrats . This would be an example of two independent samples.

Question 21

A Marine boot camp instructor recorded the time in which each of 15 recruits completed an obstacle course both before and after basic training. To test whether any improvement occurred, the instructor would use a t-distribution with 15 degrees of freedom.

Question 22

In testing the difference between two population means using two independent samples, we use the pooled variance in estimating the standard error of the sampling distribution of the sample mean difference if:

A) the sample sizes are both large.
B) the populations are normal with equal variances.
C) the populations are non-normal with unequal variances.
D) All of these choices are true.

Question 23

Two independent samples of sizes 25 and 35 are randomly selected from two normal populations with equal variances (assumed to be unknown). In order to test the difference between the population means, the test statistic is:

A) a standard normal random variable.
B) approximately standard normal random variable.
C) Student t-distributed with 58 degrees of freedom.
D) Student t-distributed with 33 degrees of freedom.

Question 24

The quantity is called the pooled variance estimate of the common variance of two unknown but equal population variances. It is the weighted average of the two sample variances, where the weights represent the:

A) sample variances.
B) sample standard deviations.
C) degrees of freedom for each sample.
D) None of these choices.

Question 25

In testing the difference between the means of two normal populations using two independent samples when the population variances are unequal, the sampling distribution of the resulting statistic is:

A) normal.
B) Student-t.
C) approximately normal.
D) approximately Student-t.

Question 26

In constructing a confidence interval estimate for the difference between the means of two independent normally distributed populations, we:

A) pool the sample variances when the unknown population variances are equal.
B) pool the sample variances when the population variances are known and equal.
C) pool the sample variances when the population means are equal.
D) never pool the sample variances.

Question 27

The number of degrees of freedom associated with the t-test, when the data are gathered from a matched pairs experiment with 8 pairs, is 7.

Question 28

Two independent samples of sizes 40 and 50 are randomly selected from two populations to test the difference between the population means . Assume the population variances are known. The sampling distribution of the sample mean difference is:

A) normally distributed.
B) approximately normal.
C) Student t-distributed with 88 degrees of freedom.
D) None of these choices.

Question 29

If we are testing for the difference between the means of two independent populations with equal variances, samples of n₁ = 15 and n₂ = 15 are taken, then the number of degrees of freedom is equal to

A) 13
B) 14
C) 28
D) 29

Question 30

In comparing two population means of interval data, we must decide whether the samples are independent (in which case the parameter of interest is ) or matched pairs (in which case the parameter is $\mu$ _D) in order to select the correct test statistic.

Question 31

When comparing two population means using data that are gathered from a matched pairs experiment, the test statistic for $\mu$ _D is Student t-distributed with v = n_D -1 degrees of freedom, provided that the differences are normally distributed.

Question 32

When testing vs. , the observed value of the z-score was found to be -2.15. Then, the p-value for this test would be

A) .0158
B) .0316
C) .9842
D) .9684

Question 33

Two independent samples of sizes 20 and 30 are randomly selected from two normally distributed populations. Assume that the population variances are unknown but equal. In order to test the difference between the population means, , the sampling distribution of the sample mean difference, , is:

A) normal.
B) Student-t with 50 degrees of freedom.
C) Student-t with 48 degrees of freedom.
D) None of these choices.

Question 34

In testing the difference between the means of two normally distributed populations, the number of degrees of freedom associated with the unequal-variances t-test statistic usually results in a non-integer number. It is recommended that you:

A) round to the nearest integer.
B) change the sample sizes until the number of degrees of freedom becomes an integer.
C) assume that the population variances are equal, and then use df = n₁ + n₂-2.
D) None of these choices.

Question 35

The expected value of the difference of two sample means equals the difference of the corresponding population means when:

A) the populations are normally distributed.
B) the samples are independent.
C) the populations are approximately normal and the sample sizes are large.
D) All of these choices are true.

Question 36

If there are 10 pairs of data in a matched pairs experiment, the degrees of freedom for the corresponding t-test is 18.

Question 37

In testing the difference between two population means for which the population variances are unknown and not assumed to be equal, two independent samples are drawn from the populations. Which of the following tests is appropriate?

A) z-test
B) pooled-variances t-test
C) unequal variances t-test
D) None of these choices.

Question 38

In testing for the differences between the means of two independent populations where the variances in each population are unknown but assumed equal, the degrees of freedom is:

A) n₁ + n₂
B) n₁ + n₂ - 2
C) n₁ + n₂ - 1
D) None of these choices

Question 39

Given the information: the number of degrees of freedom that should be used in the pooled variance t-test is:

A) 40
B) 38
C) 15
D) 25

Question 40

The t-test for the difference between the means of two independent populations assumes that the respective:

A) sample sizes are equal.
B) populations are normal.
C) means are equal.
D) All of these choices are true.

Question 41

A test is being conducted to test the difference between two population means using data that are gathered from a matched pairs experiment. If the paired differences are normal, then the distribution used for testing is the:

A) normal distribution.
B) binomial distribution.
C) Student t-distribution.
D) F-distribution.

Question 42

The symbol refers to:

A) the difference in the means of two independent populations.
B) one matched pairs difference.
C) the mean difference in the pairs of observations.
D) None of these choices.

Question 43

When the sample sizes are equal, the pooled variance of the two samples is the ____________________ of the two sample variances.

Question 44

A political analyst in Iowa surveys a random sample of registered Democrats and compares the results with those obtained from a random sample of registered Republicans. This would be an example of ____________________ samples.

Question 45

When two population variances are ____________________ we estimate each population variance with its sample variance. The test statistic of is approximately Student t-distributed with n₁ + n₂ - 2 degrees of freedom.

Question 46

The pooled variance estimator is the ____________________ average of the two sample variances.

Question 47

When the population variances are unknown and unequal, we estimate each population variance with its ____________________ variance.

Question 48

The test for the mean difference in a matched pairs design requires the differences to have a(n) ____________________ distribution.

Question 49

The equal-variances test statistic of is Student t-distributed with n₁ + n₂ - 2 degrees of freedom provided that the two populations are ____________________.

Question 50

If some natural relationship exists between each pair of observations that provides a logical reason to compare the first observation of sample 1 with the first observation of sample 2, the second observation of sample 1 with the second observation of sample 2, and so on, the samples are referred to as:

A) matched pairs.
B) independent samples.
C) nonrandom samples.
D) None of these choices.

Question 51

A political analyst in Hawaii surveys a random sample of registered Democrats and compares the results with those obtained from a random sample of registered Republicans. This would be an example of:

A) independent samples.
B) dependent samples.
C) independent samples only if the sample sizes are equal.
D) dependent samples only if the sample sizes are equal.

Question 52

The unequal-variances test statistic of has an approximate ____________________ distribution with n₁ + n₂- 2 degrees of freedom.

Question 53

If you are testing to see if a weight loss program is working, and you subtract the weights before - after for a group of 10 people, the alternative hypothesis is that the mean difference is ____________________ 0.

Question 54

____________________ samples are those for which the selection process for one is not related to the selection process for the other.

Question 55

Two measurements from the same individuals is an example of data collected from a(n) ____________________ experiment.

Question 56

In a matched pairs experiment the parameter of interest is the ____________________ of the population of ____________________.

Question 57

The number of degrees of freedom associated with the t-test, when the data are gathered from a matched pairs experiment with 10 pairs, is:

A) 10
B) 20
C) 9
D) 18

Question 58

The pooled-variances t-test is used when the two population variances are ____________________.

Question 59

The degrees of freedom for a test of the mean of the paired differences is the number of ____________________ minus ____________________.

Question 60

Suppose we randomly selected 250 people, and on the basis of their responses to a survey we assigned them to one of two groups: high-risk group and low-risk group. We then recorded the blood pressure for the members of each group. Such data are called:

A) observational.
B) experimental.
C) matched.
D) None of these choices.

Question 61

Aptitude Test Scores: Two random samples of 40 students were drawn independently from two populations of students. Assume their aptitude tests are normally distributed (total points = 100). The following statistics regarding their scores in an aptitude test were obtained: .

-Explain how to use the 95% confidence interval to test the hypotheses at $\alpha$ = .05.

Question 62

In testing the hypothesis vs. , two random samples from two normal populations produced the following statistics: . What conclusion can we draw at the 1% significance level?

Question 63

What conclusion can we draw at the 5% significance level?

Question 64

Additives: A food processor wants to compare two additives for their effects on retarding spoilage. Suppose 16 cuts of fresh meat are treated with additive A and 16 are treated with additive B, and the number of hours until spoilage begins is recorded for each of the 32 cuts of meat. The results are summarized in the table below

Assume population variances are equal. Calculate the pooled variance and the value of the test statistic.

Question 65

Undergraduates' Test Scores: 35 undergraduate students who completed two years of college were asked to take a basic mathematics test. The mean and standard deviation of their scores were 75.1 and 12.8, respectively. In a random sample of 50 students who only completed high school, the mean and standard deviation of the test scores were 72.1 and 14.6, respectively

-Estimate with 90% confidence the difference in mean scores between the two groups of students.

Question 66

Explain how to use the 95% confidence interval to test the hypotheses at $\alpha$ = .05.

Question 67

Engine Wear: To compare the wearing of two types of automobile engines, 1 and 2, an experimenter chose to "pair" the measurements, comparing the wear for the two types of engines on each of 7 automobiles, as shown below.


-Estimate with 90% confidence the mean difference and interpret.

Question 68

The service manager of a car dealer wants to determine if owners of new cars (two years old or less) tune up their cars more frequently than owners of older cars (more than two years old). From his records he takes a random sample of ten new cars and ten older cars and determines the number of times the cars were tuned up in the last 12 months. The data follow. Do these data allow the service station owner to infer at the 10% significance level that new car owners tune up their cars more frequently than older car owners?

Question 69

Additives: A food processor wants to compare two additives for their effects on retarding spoilage. Suppose 16 cuts of fresh meat are treated with additive A and 16 are treated with additive B, and the number of hours until spoilage begins is recorded for each of the 32 cuts of meat. The results are summarized in the table below

State the null and alternative hypotheses to determine if the average number of hours until spoilage begins differs for the additives A and B.

Question 70

Undergraduates' Test Scores: 35 undergraduate students who completed two years of college were asked to take a basic mathematics test. The mean and standard deviation of their scores were 75.1 and 12.8, respectively. In a random sample of 50 students who only completed high school, the mean and standard deviation of the test scores were 72.1 and 14.6, respectively

-Can we infer at the 10% significance level that a difference exists between the two groups?

Question 71

Engine Wear: To compare the wearing of two types of automobile engines, 1 and 2, an experimenter chose to "pair" the measurements, comparing the wear for the two types of engines on each of 7 automobiles, as shown below.


-Determine whether these data are sufficient to infer at the 10% significance level that the two types of engines wear differently.

Question 72

Additives: A food processor wants to compare two additives for their effects on retarding spoilage. Suppose 16 cuts of fresh meat are treated with additive A and 16 are treated with additive B, and the number of hours until spoilage begins is recorded for each of the 32 cuts of meat. The results are summarized in the table below


-Determine the rejection region at $\alpha$ = .05 and write the proper conclusion.

Question 73

If you are testing to see if a weight loss program is working, and you subtract the weights after - before for a group of 10 people, the alternative hypothesis is that the mean difference is ____________________ 0.

Question 74

Promotional Campaigns: The general manager of a chain of fast food chicken restaurants wants to determine how effective their promotional campaigns are. In these campaigns "20% off" coupons are widely distributed. These coupons are only valid for one week. To examine their effectiveness, the executive records the daily gross sales (in $1,000s) in one restaurant during the campaign and during the week after the campaign ends. The data is shown below.


-Can they infer at the 5% significance level that sales increase during the campaign?

Question 75

Aptitude Test Scores: Two random samples of 40 students were drawn independently from two populations of students. Assume their aptitude tests are normally distributed (total points = 100). The following statistics regarding their scores in an aptitude test were obtained: .
Test at the 5% significance level to determine whether we can infer that the two population means differ.

Question 76

Clothing Expenditures: A marketing consultant was in the process of studying the perceptions of married couples concerning their monthly clothing expenditures. He believed that the husband's perception would be higher than the wife's. To judge his belief, he takes a random sample of ten married couples and asks each spouse to estimate the family clothing expenditure (in dollars) during the previous month. The data are shown below.


-Can the consultant conclude at the 5% significance level that the husband's estimate is higher than the wife's estimate?

Question 77

Aptitude Test Scores: Two random samples of 40 students were drawn independently from two populations of students. Assume their aptitude tests are normally distributed (total points = 100). The following statistics regarding their scores in an aptitude test were obtained: .

-Estimate with 95% confidence the difference between the two population means.

Question 78

Estimate with 95% confidence the difference between the two population means.

Question 79

Promotional Campaigns: The general manager of a chain of fast food chicken restaurants wants to determine how effective their promotional campaigns are. In these campaigns "20% off" coupons are widely distributed. These coupons are only valid for one week. To examine their effectiveness, the executive records the daily gross sales (in $1,000s) in one restaurant during the campaign and during the week after the campaign ends. The data is shown below.


-Estimate with 95% confidence the mean difference and interpret.

Question 80

Motorcycle insurance appraisers examine motorcycles that have been involved in accidental collisions to assess the cost of repairs. An insurance executive is concerned that different appraisers produce significantly different assessments. In an experiment 10 motorcycles that have recently been involved in accidents were shown to two appraisers. Each assessed the estimated repair costs. These results are shown below. Can the executive conclude at the 5% significance level that the appraisers differ in their assessments?