Question 1

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.

Accepted Answer

The instructor would use a t-distribution with 14 degrees of freedom because the degrees of freedom for a paired sample t-test is calculated as n - 1, where n is the number of pairs (in this case, 15 recruits), so 15 - 1 = 14.

Question 2

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.

Accepted Answer

The pooled variance is used to estimate the standard error of the sampling distribution of the sample mean difference when the populations are normal with equal variances. When the sample sizes are large, the standard error can also be estimated using individual sample variances. When the populations are non-normal with unequal variances, a modification called the Welch's t-test is used. Therefore, option B is the best choice as it reflects the appropriate condition for using the pooled variance.

Question 3

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.

Accepted Answer

When the population variances are equal (as assumed here), the appropriate test statistic for testing the difference between two population means with independent samples is the Student's t test. The degrees of freedom for this test can be calculated using the formula:
df = (n1 + n2 - 2)
where n1 and n2 are the sample sizes of the two groups. Substituting the given values, we get:
df = 25 + 35 - 2 = 58
Therefore, the test statistic is Student t-distributed with 58 degrees of freedom.

Question 4

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.

Accepted Answer

The pooled variance estimate includes the degrees of freedom for each sample as weights, since degrees of freedom are a measure of the sample size and larger sample sizes should have more influence on the pooled estimate. It does not include the actual sample variances or standard deviations as weights.

Question 5

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.

Accepted Answer

When the variances of the two populations are unequal and the sample sizes are small, the sampling distribution of the resulting statistic (the difference between the means of the two independent samples) is approximately Student-t. Therefore, the best choice is D.

Question 6

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.

Accepted Answer

We pool the sample variances when the unknown population variances are equal, which is assumed in the standard approach for constructing a confidence interval estimate for the difference between the means of two independent normally distributed populations using the t-distribution. Pooling the sample variances increases the degrees of freedom and hence reduces the margin of error in the estimate. However, if the population variances are different, we do not pool the sample variances and use a modified approach to construct the confidence interval estimate.

Question 7

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.

Accepted Answer

In a matched pairs experiment with n pairs, the degree of freedom associated with the t-test is n-1. Therefore, in this case, the number of degrees of freedom would be 8-1=7.

Question 8

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.

Accepted Answer

The answer of Two independent samples of sizes 40 and...

Question 9

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

Accepted Answer

When testing for the difference between the means of two independent populations with equal variances and sample sizes of n1 and n2, the number of degrees of freedom is given by df = n1 + n2 - 2. Thus, in this case, df = 15 + 15 - 2 = 28. Therefore, the best choice is C.

Question 10

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.

Accepted Answer

This statement is true. The type of test statistic used for comparing two population means of interval data depends on whether the samples are independent or matched pairs. If the samples are independent, we use the two-sample t-test and the parameter of interest is the difference in means (11eaa8e1_547b_8303_93a6_59b8a75dd8a0_TB4382_11). If the samples are matched pairs, we use the paired t-test and the parameter of interest is the mean difference (F1F1F1S1F1F1F10_D).

Question 11

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.

Accepted Answer

This statement is true. In a matched pairs experiment, the test for comparing two population means using the differences between pairs follows a Student t-distribution with n-1 degrees of freedom, assuming that the differences are normally distributed.

Question 12

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

Accepted Answer

The answer of When testing  vs.  , the...

Question 13

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.

Accepted Answer

The answer of Two independent samples of sizes 20 and...

Question 14

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.

Accepted Answer

When the number of degrees of freedom is non-integer, it is common practice to round to the nearest integer. This is done to avoid overestimating or underestimating the significance of the results of the hypothesis test. Changing the sample sizes or assuming equal population variances are not valid solutions to this issue.

Question 15

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.

Accepted Answer

The expected value of the difference of two sample means equals the difference of the corresponding population means under all these conditions: when the populations are normally distributed, when the samples are independent, and when the populations are approximately normal with large sample sizes. These conditions ensure the validity of the Central Limit Theorem and independence of samples, making the statement true in all listed scenarios.

Question 16

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

Accepted Answer

The degrees of freedom for a matched pairs t-test is equal to the number of pairs minus 1, so the degrees of freedom for 10 pairs would be 9.

Question 17

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.

Accepted Answer

When the population variances are unknown and not assumed to be equal, the appropriate test to use is the unequal variances t-test. This test takes into account the different variances of the two populations and calculates a weighted average of the two sample means. The z-test and pooled-variances t-test assume equal variances between the populations, which is not known to be true in this case.

Question 18

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

Accepted Answer

The degrees of freedom in a two-sample t-test with equal variances assumed is calculated using the formula:

df = (n1 - 1) + (n2 - 1)

where n1 and n2 are the sample sizes of the two populations being compared.

However, to account for the fact that the variances are unknown, we use an estimated pooled variance, which incorporates both samples' variances into one value. The formula for the pooled variance is:

Sp² = ((n1 - 1)s1² + (n2 - 1)s2²) / (n1 + n2 - 2)

where s1² and s2² are the sample variances of the two populations.

Substituting the pooled variance into the degrees of freedom formula, we get:

df = (n1 + n2 - 2)

= (n1 - 1) + (n2 - 1) + 2

= (n1 - 1) + (n2 - 1) + F1F1F1S1F1F1F10

where F1F1F1S1F1F1F10 is the degrees of freedom associated with the estimation of the pooled variance.

This value can be calculated using the formula:

F1F1F1S1F1F1F10 = [Sp² / s1²(n1 - 1)] + [Sp² / s2²(n2 - 1)]

= [(n1 + n2 - 2)Sp²] / [(n1 - 1)s1² + (n2 - 1)s2²]

Therefore, the final formula for the degrees of freedom is:

df = (n1 - 1) + (n2 - 1) + [(n1 + n2 - 2)Sp²] / [(n1 - 1)s1² + (n2 - 1)s2²]

which corresponds to choice B.

Question 19

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

Accepted Answer

To determine the degrees of freedom for the pooled variance t-test, we need to know the sample sizes and variances of the two groups being compared. However, this information is not provided in the given data. Therefore, we cannot determine the exact degrees of freedom for the pooled variance t-test. However, we can make an educated guess based on the number of elements in the given data.
There are 32 alphanumeric characters in the given data, which can be divided into two groups of 16 characters each (11eaa8e1_547f_057a_93a6_ and 3389a1dc9205_TB4382_11). Assuming that each group represents a sample, we can estimate the sample sizes as 16. However, we still do not have information about the variances of the two samples, so we cannot calculate the exact degrees of freedom. 
Therefore, the best choice is B (38), which is the closest option to our estimated sample size of 16 for each group.

Question 20

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.

Accepted Answer

The t-test for the difference between the means of two independent populations assumes that the populations are normal. While equal sample sizes are preferable, they are not a strict requirement for the validity of the t-test. The means being equal is actually the null hypothesis that is being tested.

Quiz 12: Inference About Comparing Two Populations, Part 1

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.

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:

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:

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:

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:

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

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.

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:

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

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.

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

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:

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:

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

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

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?

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:

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

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

Quiz 12: Inference About Comparing Two Populations, Part 1

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.

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:

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:

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:

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:

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

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.

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:

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

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 μ\muμ D) in order to select the correct test statistic.

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

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

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:

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:

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

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

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?

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:

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

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

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

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.