Question 1

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

The appropriate test is the unequal variances t-test, also known as the Welch's test. This test does not assume that the variances of the two populations are equal, making it a good choice when dealing with independent samples from populations with unknown variances. The z-test requires the population variances to be known or assumed to be equal, and the pooled-variances t-test assumes equal variances, which may not hold in some cases. Therefore, the unequal variances t-test is the best choice in this scenario.

Question 2

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 for testing the difference between the means of two independent populations with equal but unknown variances is given by:

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

where n1 and n2 are the sample sizes of the two populations. However, when the variances are assumed to be equal, we can pool the sample variances and estimate a common variance, given by:

s2 = [(n1 - 1)s1^2 + (n2 - 1)s2^2] / (n1 + n2 - 2)

The test statistic for this scenario is given by:

t = (x1 - x2) / [s * sqrt(1/n1 + 1/n2)]

where x1 and x2 are the sample means, s is the pooled estimate of the common variance, and n1 and n2 are the sample sizes.

Substituting the expression for s, we get:

t = (x1 - x2) / [sqrt{[(n1 - 1)s1^2 + (n2 - 1)s2^2] / (n1 + n2 - 2)} * sqrt(1/n1 + 1/n2)]

The degrees of freedom for the t-distribution is given by:

df = (s1^2/n1 + s2^2/n2)^2 / {[(s1^2/n1)^2 / (n1 - 1)] + [(s2^2/n2)^2 / (n2 - 1)]}

Simplifying this expression, we get:

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

Therefore, the correct choice is B, which gives the formula for the degrees of freedom as:

df = n1 + n2 - 2 / [(n1 - 1)s1^2 + (n2 - 1)s2^2] / (n1 + n2 - 2) * (1/n1 + 1/n2)

Question 3

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 4

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.

Accepted Answer

The answer of When two population variances are ____________________ we...

Question 5

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.

Accepted Answer

The answer of A political analyst in Iowa surveys a...

Question 6

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

Accepted Answer

The answer of ____________________ samples are those for which the...

Question 7

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 8

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

Accepted Answer

The answer of The unequal-variances test statistic of  has...

Question 9

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.

Accepted Answer

The two samples (registered Democrats and Republicans) are independent of each other as they are selected randomly and do not depend on each other. Therefore, this is an example of independent samples. The sample sizes being equal or not does not affect whether the samples are independent or dependent.

Question 10

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

When the unknown population variances are equal, we can pool the sample variances to increase the degrees of freedom and hence, the precision of the confidence interval estimate.

Question 11

In testing for differences between the means of two independent populations the null hypothesis is:
A) 
B) 
C) 
D)

Accepted Answer

The null hypothesis in this scenario is typically that the means of the two populations being tested are equal. Choice B, "11ec8e60_33be_0b04_bce4_ab5fe5ada8bd_TB34225555_11," is the only option that represents this null hypothesis.

Question 12

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.

Accepted Answer

The data in this scenario is observational since the groups were not randomly assigned or manipulated by the researchers. The researchers simply observed naturally occurring groups (high-risk vs low-risk) and then recorded their corresponding blood pressure measurements.

Question 13

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 variances are unknown and assumed to be equal, the appropriate test statistic is the Student's t-distribution with the degrees of freedom calculated using the formula: df = n1 + n2 - 2 = 25 + 35 - 2 = 58. Therefore, the correct choice is C, which indicates that the test statistic is Student t-distributed with 58 degrees of freedom.

Question 14

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

The number of degrees of freedom equals n1 + n2 - 2, where n1 and n2 are the sample sizes of the two groups being compared. 
In this case, there is only one group being compared, so the number of degrees of freedom should be equal to the sample size minus one. 
There are 11 alphanumeric characters in the given information, so the sample size is 11. Therefore, the number of degrees of freedom is 11 - 1 = 10. 
However, since this is a pooled variance t-test, we need to use the degrees of freedom for the pooled variance, which is (n1+n2-2). Since there is only one group being compared, the pooled variance is equal to the variance of the sample, and the degrees of freedom should be equal to the sample size minus one, which is 11 - 1 = 10. 
Therefore, using the given information, the best choice is B.

Question 15

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

When 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. If the sample sizes are both large, we can also use the pooled variance. If the populations are non-normal with unequal variances, we do not use the pooled variance.

Question 16

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 population variances are unequal and the sample size is small (less than 30), the sampling distribution of the resulting statistic is approximately Student-t.

Question 17

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

Accepted Answer

The answer of The equal-variances test statistic of  is...

Question 18

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

Accepted Answer

The answer of When the sample sizes are equal, the...

Question 19

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

The number of degrees of freedom is given by the formula df = n1 + n2 - 2, where n1 and n2 are the sample sizes of the two populations. Substituting n1 = 15 and n2 = 15 in the formula, we get df = 28. Therefore, the correct choice is (C).

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 having equal sample sizes and means can also be helpful, they are not strictly necessary assumptions for conducting the t-test.

Quiz 13: Inference About Comparing Two Populations

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:

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:

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.

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.

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

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

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

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:

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

In testing for differences between the means of two independent populations the null hypothesis is:

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:

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:

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

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:

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:

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

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

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

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

Quiz 13: Inference About Comparing Two Populations

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:

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:

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.

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.

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

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

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

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:

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

In testing for differences between the means of two independent populations the null hypothesis is:

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:

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:

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

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:

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:

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

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

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

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

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.

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

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