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Chapter 11

Two-Sample Tests of Hypothesis

True / False Questions

1. If the null hypothesis states that there is no difference between the mean income of males and the mean income of females, then the test is one-tailed.

True False

2. If the null hypothesis states that there is no difference between the mean net income of retail stores in Chicago and New York City, then the test is two-tailed.

True False

3. When the standard deviations are equal but unknown, a test for the differences between two population means has n 1 degrees of freedom.

True False

4. If we are testing for the difference between two population means and assume that the two populations have equal but unknown standard deviations, the variances are pooled to compute the best estimated variance.

True False

5. If we are testing for the difference between two population means and assume that the two populations have equal and unknown standard deviations, the degrees of freedom are computed as (n1)(n2) 1.

True False

6. A statistics professor wants to compare grades in two different classes of the same course. This is an example of a paired sample.

True False

7. If we are testing for the difference between two population means, it is assumed that the sample observations from one population are independent of the sample observations from the other population.

True False

8. When dependent samples are used to test for differences in the means, we compute paired differences.

True False

9. If two dependent samples of size 20 are used to test the difference between the means, the degrees of freedom for a t-statistic are 19.

True False

10. When dependent samples are used to test for differences in the means, we pool the sample variances.

True False

Multiple Choice Questions

11. A recent study focused on the number of times men and women send a Twitter message in a day. The sample information is summarized below.

At the .01 significance level, is there a difference in the mean number of times men and women send a Twitter message in a day? What is the test statistic for this hypothesis?

A. z-statistic

B. t-statistic

C. p-statistic

D. df-statistic

12. A recent study focused on the number of times men and women send a Twitter message in a day. The information is summarized next.

At the .01 significance level, is there a difference in the mean number of times men and women send a Twitter message in a day? What is the value of the test statistic for this hypothesis test?

A. 2.668

B. 2.672

C. 2.58

D. 2.40

13. A recent study focused on the number of times men and women send a Twitter message in a day. The information is summarized next.

At the .01 significance level, is there a difference in the mean number of times men and women send a Twitter message in a day? What is the p-value for this hypothesis test?

A. 0.0500

B. 0.0164

C. 0.0001

D. 0.0082

14. A recent study focused on the number of times men and women send a Twitter message in a day. The information is summarized next.

At the .01 significance level, is there a difference in the mean number of times men and women send a Twitter message in a day? Based on the p-value, what is your conclusion?

A. Reject the null hypothesis and conclude the means are different.

B. Reject the null hypothesis and conclude the means are the same.

C. Fail to reject the null hypothesis.

D. Fail to reject the null hypothesis and conclude the means are different.

15. A recent study focused on the amount of money single men and women save monthly. The information is summarized next. Assume that the population standard deviations are equal.

At the .01 significance level, do women save more money than men? What is the test statistic for this hypothesis?

A. z-statistic

B. t-statistic

C. p-statistic

D. df-statistic

16. A recent study focused on the amount of money single men and women save monthly. The information is summarized next. Assume that the population standard deviations are equal.

At the .01 significance level, do women save more money than men? What is the critical value for this hypothesis test?

A. +6.213

B. +2.369

C. +2.632

D. +2.40

17. A recent study focused on the amount of money single men and women save monthly. The information is summarized next. Assume that the population standard deviations are equal.

At the .01 significance level, do women save more money than men? What is the value of the test statistic for this hypothesis test?

A. +6.213

B. +1.318

C. +2.632

D. +2.40

18. A recent study focused on the amount of money single men and women save monthly. The information is summarized next. Assume that the population standard deviations are equal.

At the .01 significance level, what is the conclusion about the way women and men save?

A. Reject the null hypothesis and conclude that women save more than men.

B. Reject the null hypothesis and conclude that women and men save the same amount.

C. Fail to reject the null hypothesis.

D. Fail to reject the null hypothesis and conclude the means are different.

19. If the null hypothesis that two means are equal is true, where will 97% of the computed z values lie between?

A. 2.58

B. 2.33

C. 2.17

D. 2.07

20. Assuming the population variances are known, the population variance of the difference between two means is _____________.

A. The sum of the two means

B. The sum of the two population variances

C. The sum of the two population standard deviations

D. The sum of the two sample sizes for each population

21. When testing the difference between two population means, the sample variances are pooled to estimate the population variance when ________________.

A. The population variances are known and equal

B. The population means are known

C. The population variances are assumed unequal and unknown

D. The population variances are assumed equal but unknown

22. When testing the difference between two dependent population means, the test statistic is based on a ______________.

A. Pooled variance

B. Standard deviation of the differences

C. Pooled proportion

D. Sum of the population variances

23. The net weights (in grams) of a sample of bottles filled by a machine manufactured by Edne, and the net weights of a sample filled by a similar machine manufactured by Orno, Inc., are:

Testing the claim at the 0.05 level that the mean weight of the bottles filled by the Orno machine is greater than the mean weight of the bottles filled by the Edne machine, what is the critical value? Assume equal standard deviations for both samples.

A. +2.179

B. +2.145

C. +1.782

D. +1.761

24. Which condition must be met to conduct a test for the difference in two sample means using a z-statistic?

A. The data must be at least of nominal scale.

B. The populations must be normal.

C. The two population standard deviations must be known.

D. The samples are dependent.

25. When is it appropriate to use the paired difference t-test?

A. When four samples are compared at once

B. When any two samples are compared

C. When two independent samples are compared

D. When two dependent samples are compared

26. We test for a hypothesized difference between two population means: H0: 1 = 2. The population standard deviations are unknown but assumed equal. The number of observations in the first sample is 15, and 12 in the second sample. How many degrees of freedom are associated with the critical value?

A. 24

B. 25

C. 26

D. 27

27. For a hypothesis comparing two population means, H0: 1 2, what is the critical value for a one-tailed hypothesis test, using a 5% significance level, with both sample sizes equal to 13? Assume the population standard deviations are equal.

A. 1.711

B. +1.711

C. +2.060

D. +2.064

28. For a hypothesis test comparing two population means, the combined degrees of freedom are 24. Which of the following statements about the two sample sizes is NOT true? Assume the population standard deviations are equal.

A. n1 = 11; n2 = 13

B. n1 = 12; n2 = 14

C. n1 = 13; n2 = 13

D. n1 = 10; n2 = 16

29. Two samples, one of size 14 and the second of size 13, are selected to test the difference between two population means. How many degrees of freedom are used to find the critical value? Assume the population standard deviations are equal.

A. 27

B. 26

C. 25

D. 14

30. Twenty randomly selected statistics students were given 15 multiple-choice questions and 15 open-ended questions, all on the same material. The professor was interested in determining if students scored higher on the multiple-choice questions. This experiment is an example of ________________.

A. A one-sample test of means

B. A two-sample test of means

C. A paired t-test

D. A test of proportions

31. A national manufacturer of ball bearings is experimenting with two different processes for producing precision ball bearings. It is important that the diameters be as close as possible to an industry standard. The output from each process is sampled and the average error from the industry standard is measured in millimeters. The results are presented next.

The researcher is interested in determining whether there is evidence that the two processes yield different average errors. The population standard deviations are unknown but are assumed equal. What is the null hypothesis?

A. H0: A = B

B. H0: A B

C. H0: A B

D. H0: A > B

32. A national manufacturer of ball bearings is experimenting with two different processes for producing precision ball bearings. It is important that the diameters be as close as possible to an industry standard. The output from each process is sampled and the average error from the industry standard is measured in millimeters. The results are presented next.

The researcher is interested in determining whether there is evidence that the two processes yield different average errors. The population standard deviations are unknown but assumed equal. What is the alternate hypothesis?

A. H1: A = B

B. H1: A B

C. H1: A B

D. H1: A > B

33. A national manufacturer of ball bearings is experimenting with two different processes for producing precision ball bearings. It is important that the diameters be as close as possible to an industry standard. The output from each process is sampled and the average error from the industry standard is measured in millimeters. The results are presented next.

The researcher is interested in determining whether there is evidence that the two processes yield different average errors. The population standard deviations are unknown but are assumed equal. What are the degrees of freedom?

A. 10

B. 13

C. 26

D. 24

34. A national manufacturer of ball bearings is experimenting with two different processes for producing precision ball bearings. It is important that the diameters be as close as possible to an industry standard. The output from each process is sampled and the average error from the industry standard is measured in millimeters. The results are presented next.

The researcher is interested in determining whether there is evidence that the two processes yield different average errors. The population standard deviations are unknown but assumed equal. What is the critical t value at the 1% level of significance?

A. +2.797

B. -2.492

C. 1.711

D. 2.797

35. A national manufacturer of ball bearings is experimenting with two different processes for producing precision ball bearings. It is important that the diameters be as close as possible to an industry standard. The output from each process is sampled and the average error from the industry standard is measured in millimeters. The results are presented next.

The researcher is interested in determining whether there is evidence that the two processes yield different average errors. The population standard deviations are unknown but are assumed equal. What is the computed value of t?

A. +2.797

B. -1.000

C. -3.299

D. 0.5938

36. A national manufacturer of ball bearings is experimenting with two different processes for producing precision ball bearings. It is important that the diameters be as close as possible to an industry standard. The output from each process is sampled and the average error from the industry standard is measured in millimeters. The results are presented next.

The researcher is interested in determining whether there is evidence that the two processes yield different average errors. The population standard deviations are unknown but are assumed equal. If we test the null hypothesis at the 1% level of significance, what is the decision?

A. Reject the null hypothesis and conclude the means are different.

B. Reject the null hypothesis and conclude the means are the same.

C. Fail to reject the null hypothesis.

D. Fail to reject the null hypothesis and conclude the means are different.

37. A national manufacturer of ball bearings is experimenting with two different processes for producing precision ball bearings. It is important that the diameters be as close as possible to an industry standard. The output from each process is sampled and the average error from the industry standard is measured in millimeters. The results are presented next.

The researcher is interested in determining whether there is evidence that the two processes yield different average errors. The population standard deviations are unknown but are assumed equal. This example is what type of test?

A. A one-sample test of means

B. A two-sample test of means

C. A paired t-test

D. A test of proportions

38. The results of a mathematics placement exam at two different campuses of Mercy College follow:

What is the null hypothesis if we want to test the hypothesis that the mean score on Campus 1 is higher than on Campus 2?

A. H0: 1 = 0

B. H0: 2 = 0

C. H0: 1 = 2

D. H0: 1 2

39. The results of a mathematics placement exam at two different campuses of Mercy College follow:

What is the alternative hypothesis if we want to test the hypothesis that the mean score on Campus 1 is higher than on Campus 2?

A. H1: 1 = 0

B. H1: 2 = 0

C. H1: 1 > 2

D. H1: 1 2

40. The results of a mathematics placement exam at two different campuses of Mercy College follow:

What is the computed value of the test statistic?

A. 9.30

B. 2.60

C. 3.37

D. 3.40

41. The results of a mathematics placement exam at two different campuses of Mercy College follow:

Given that the two population standard deviations are known, what is the p-value?

A. 1.0

B. 0.0

C. 0.05

D. 0.95

42. Accounting procedures allow a business to evaluate their inventory costs based on two methods: LIFO (Last In First Out) or FIFO (First In First Out). A manufacturer evaluated its finished goods inventory (in $000s) for five products with the LIFO and FIFO methods. To analyze the difference, they computed (FIFO LIFO) for each product. Based on the following results, does the LIFO method result in a lower cost of inventory than the FIFO method?

What is the null hypothesis?

A. H0: d = 0

B. H0: d 0

C. H0: d 0

D. H0: d 0

43. Accounting procedures allow a business to evaluate their inventory costs based on two methods: LIFO (Last In First Out) or FIFO (First In First Out). A manufacturer evaluated its finished goods inventory (in $000s) for five products with the LIFO and FIFO methods. To analyze the difference, they computed (FIFO LIFO) for each product. Based on the following results, does the LIFO method result in a lower cost of inventory than the FIFO method?

What is the alternate hypothesis?

A. H1: d = 0

B. H1: d 0

C. H1: d 0

D. H1: d > 0

44. An investigation of the effectiveness of a training program to improve customer relationships included a pre-training and post-training customer survey. To compare the differences, they computed (post-training survey score pre-training survey score). Seven customers were randomly selected and completed both surveys. The results follow.

This analysis is an example of _____________.

A. A one-sample test of means

B. A two-sample test of means

C. A paired t-test

D. A test of proportions

45. An investigation of the effectiveness of a training program to improve customer relationships included a pre-training and post-training customer survey. To compare the differences they computed (post-training survey score pre-training survey score). Seven customers were randomly selected and completed both surveys. The results follow.

For a 0.05 significance level, what is the critical value?

A. 1.943

B. 1.895

C. 1.645

D. 2.447

46. An investigation of the effectiveness of a training program to improve customer relationships included a pre-training and post-training customer survey. To compare the differences they computed (post-training survey score pre-training survey score). Seven customers were randomly selected and completed both surveys. The results follow.

What is the value of the test statistic?

A. 1.943

B. 1.895

C. 2.542

D. 2.447

47. An investigation of the effectiveness of a training program to improve customer relationships included a pre-training and post-training customer survey. To compare the differences they computed (post-training survey score pre-training survey score). Seven customers were randomly selected and completed both surveys. The results follow.

For a 0.05 significance level, what is the decision regarding the hypothesis that the training was effective in improving customer relationships?

A. Reject the null hypothesis and conclude that the training was effective.

B. Reject the null hypothesis and conclude that the training was ineffective.

C. Fail to reject the null hypothesis and conclude that the mean survey scores are the same.

D. Fail to reject the null hypothesis and conclude that the mean survey scores are not equal.

48. A company is researching the effectiveness of a new website design to decrease the time to access a website. Five website users were randomly selected, and their times (in seconds) to access the website with the old and new designs were recorded. To compare the times, they computed (new website design time old website design time). The results follow.

What is the null hypothesis?

A. H0: d = 0

B. H0: d 0

C. H0: d 0

D. H0: d 0

49. A company is researching the effectiveness of a new website design to decrease the time to access a website. Five website users were randomly selected, and their times (in seconds) to access the website with the old and new designs were recorded. To compare the times, they computed (new website design time old website design time). The results follow.

What is the alternative hypothesis?

A. H1: d = 0

B. H1: d 0

C. H1: d > 0

D. H1: d < 0 50. A company is researching the effectiveness of a new website design to decrease the time to access a website. Five website users were randomly selected, and their times (in seconds) to access the website with the old and new designs were recorded. To compare the times, they computed (new website design time - old website design time). The results follow. What is the value of the test statistic? A. 2.256 B. 1.895 C. 3.747 D. 2.447 51. A company is researching the effectiveness of a new website design to decrease the time to access a website. Five website users were randomly selected, and their times (in seconds) to access the website with the old and new designs were recorded. To compare the times, they computed (new website design time - old website design time). The results follow. For a 0.01 significance level, what is the critical value? A. 2.256 B. 1.895 C. 3.747 D. 2.447 52. A company is researching the effectiveness of a new website design to decrease the time to access a website. Five website users were randomly selected, and their times (in seconds) to access the website with the old and new designs were recorded. To compare the times, they computed (new website design time - old website design time). The results follow. For a 0.01 significance level, what is the decision regarding the hypothesis that the training was effective in improving customer relationships? A. Reject the null hypothesis and conclude that the new design reduced the mean access times. B. Reject the null hypothesis and conclude that the new design did not reduce the mean access times. C. Fail to reject the null hypothesis. D. Fail to reject the null hypothesis and conclude that the mean access times are inaccurate. 53. Accounting procedures allow a business to evaluate their inventory costs based on two methods: LIFO (Last In First Out) or FIFO (First In First Out). A manufacturer evaluated its finished goods inventory (in $000s) for five products with the LIFO and FIFO methods. To analyze the difference, they computed (FIFO - LIFO) for each product. Based on the following results, does the LIFO method result in a lower cost of inventory than the FIFO method? What are the degrees of freedom? A. 4 B. 5 C. 15 D. 10 54. Accounting procedures allow a business to evaluate their inventory costs based on two methods: LIFO (Last In First Out) or FIFO (First In First Out). A manufacturer evaluated its finished goods inventory (in $000s) for five products with the LIFO and FIFO methods. To analyze the difference, they computed (FIFO - LIFO) for each product. Based on the following results, does the LIFO method result in a lower cost of inventory than the FIFO method? If you use the 5% level of significance, what is the critical t value? A. +2.132 B. 2.132 C. +2.262 D. 2.228 55. Accounting procedures allow a business to evaluate its inventory costs based on two methods: LIFO (Last In First Out) or FIFO (First In First Out). A manufacturer evaluated its finished goods inventory (in $000s) for five products with the LIFO and FIFO methods. To analyze the difference, they computed (FIFO - LIFO) for each product. Based on the following results, does the LIFO method result in a lower cost of inventory than the FIFO method? What is the value of calculated t? A. +0.933 B. 2.776 C. +0.47 D. -2.028 56. Accounting procedures allow a business to evaluate its inventory costs based on two methods: LIFO (Last In First Out) or FIFO (First In First Out). A manufacturer evaluated its finished goods inventory (in $000s) for five products with the LIFO and FIFO methods. To analyze the difference, they computed (FIFO - LIFO) for each product. Based on the following results, does the LIFO method result in a lower cost of inventory than the FIFO method? What is the decision at the 5% level of significance? A. Fail to reject the null hypothesis and conclude LIFO is more effective. B. Reject the null hypothesis and conclude LIFO is more effective. C. Reject the alternate hypothesis and conclude LIFO is more effective. D. Fail to reject the null hypothesis. 57. Accounting procedures allow a business to evaluate its inventory costs based on two methods: LIFO (Last In First Out) or FIFO (First In First Out). A manufacturer evaluated its finished goods inventory (in $000s) for five products with the LIFO and FIFO methods. To analyze the difference, they computed (FIFO - LIFO) for each product. Based on the following results, does the LIFO method result in a lower cost of inventory than the FIFO method? This example is what type of test? A. A one-sample test of means. B. A two-sample test of means. C. A paired t-test. D. A test of proportions. 58. When testing the hypothesized equality of two population means, the implied null hypothesis is ____________. A. H0: 1 = 0 B. H0: 1 - 2 = 0 C. H0: 2 = 0 D. H0: 1 - 2 0 59. Consider independent simple random samples that are taken to test the difference between the means of two populations. The variances of the populations are unknown, but are assumed to be equal. The sample sizes of each population are n1 = 37 and n2 = 45. The appropriate distribution to use is the: A. t distribution with df = 82. B. t distribution with df = 81. C. t distribution with df = 41. D. t distribution with df = 80. 60. Use the following table to determine whether or not there is a significant difference between the average hourly wages at two manufacturing companies. What is the point estimate of the difference between the means? A. 15 B. 0.4 C. 0.8 D. -9 61. Use the following table to determine whether or not there is a significant difference between the average hourly wages at two manufacturing companies. The p-value is ______. A. 0.036 B. 0.0336 C. 0.4664 D. 2.58 62. The following table shows sample salary information for employees with bachelor's and associate degrees for a large company in the Southeast United States. The point estimate of the difference between the means of the two populations is ______. A. 32 B. 9 C. -4.5 D. 4.5 Fill in the Blank Questions 63. When the population standard deviations are unknown, the purpose of pooling the sample variances when testing the difference between two populations is to ___________. ________________________________________ 64. If we are testing for the difference between two population means and assume that the two populations have equal and unknown standard deviations, the standard deviations are ______ to compute a point estimate of the population variance. ________________________________________ 65. When independent samples with unknown but equal standard deviations are used to test for differences in the means, we pool the sample _________________. ________________________________________ 66. The paired t test is especially appropriate when the two samples are ________. ________________________________________ 67. In one class, a statistics professor wants to compare grades on the first and second exams. This is an example of ________________ observations. ________________________________________ 68. If samples taken from two populations are dependent, then a test of ______ differences is applied. ________________________________________ 69. The paired difference test has ___________ degrees of freedom. ________________________________________ 70. When testing for a difference between the means of two dependent samples, n1 and n2 are ________________. ________________________________________ 71. When testing the null hypothesis that two population means are equal, the hypothesized difference between the population means is ________________. ________________________________________ 72. When testing for a difference between the means of two dependent samples, the sample test statistic is a ______________. ________________________________________ 73. If the null hypothesis states that there is no difference between the two population means, then the test is ______________. ________________________________________ 74. If a hypothesis states that one population mean is greater than another population mean, then the test is _______________. ________________________________________ 75. When the population standard deviations are equal but unknown, a test for the differences between two populations assumes that the populations are _____________ distributed. ________________________________________ 76. A statistics professor wants to compare grades in two different classes of the same course. This is an example of _________ populations. ________________________________________ 77. If we are testing for the difference between two population proportions, it is assumed that the sample observations from one population are _____________ of the sample observations from the other population. ________________________________________ 78. When dependent samples are used to test for differences in the means, we compute ________________ differences. ________________________________________ 79. If two dependent samples of size 100 are used to test the difference between the means, the degrees of freedom for a t-statistic are _____. ________________________________________ 80. When independent samples are used to test for differences in the population means with equal but unknown population standard deviations, we _______ these sample standard deviations. ________________________________________ Short Answer Questions 81. If we are testing for the difference between two population means and assume that the two populations have equal but unknown standard deviations, then the test has ________ degrees of freedom. 82. A study by a bank compared the mean savings of customers who were depositors for three years or less with those who had been depositors for more than three years. The results of a sample are as follows. Assuming that the financial officer wants to show that there is a difference in the average savings balance between the two classes of depositors, the null hypothesis is ____________. Essay Questions 83. What is the critical value of t for a two-tail test of the difference between two means, a level of significance of 0.10 and sample sizes of 7 and 15? Assume equal population standard deviations. 84. What is the critical value of t for the claim that the difference of two means is less than zero with = 0.025 and sample sizes of nine and seven? Assume equal population standard deviations. 85. A study by a bank compared the average savings of customers who were depositors for three years or less with those who had been depositors for more than three years. The results of a sample are: To test that the two groups of customers have equal savings rates, what is the critical value of z using = 0.05? 86. A study by a bank compared the average savings of customers who were depositors for three years or less, with those who had been depositors for more than three years. The results of a sample are: What is the computed test statistic? Round to two decimal places. 87. A study by a bank compared the average savings of customers who were depositors for three years or less with those who had been depositors for more than three years. The results of a sample are: To test that the two groups of customers have equal savings rates, what is the p-value for a two-tailed test? 88. A financial planner wants to compare the yield of income and growth mutual funds. Fifty thousand dollars is invested in each of a sample of 35 income and 40 growth funds. The mean increase for a two-year period for the income funds is $900. For the growth funds, the mean increase is $875. Income funds have a sample standard deviation of $35; growth funds have a sample standard deviation of $45. Assume that the population standard deviations are equal. At the 0.05 significance level, is there a difference in the mean yields of the two funds? What is the null hypothesis? 89. A financial planner wants to compare the yield of income and growth mutual funds. Fifty thousand dollars is invested in each of a sample of 35 income and 40 growth funds. The mean increase for a two-year period for the income funds is $900. For the growth funds, the mean increase is $875. Income funds have a sample standard deviation of $35; growth funds have a sample standard deviation of $45. Assume that the population standard deviations are equal. At the 0.05 significance level, is there a difference in the mean yields of the two funds? What is the computed value of the test statistic? 90. A financial planner wants to compare the yield of income and growth mutual funds. Fifty thousand dollars is invested in each of a sample of 35 income and 40 growth funds. The mean increase for a two-year period for the income funds is $900. For the growth funds, the mean increase is $875. Income funds have a sample standard deviation of $35; growth funds have a sample standard deviation of $45. Assume that the population standard deviations are equal. At the 0.05 significance level, is there a difference in the mean yields of the two funds? What is the p-value for the computed test statistic? 91. A financial planner wants to compare the yield of income and growth mutual funds. Fifty thousand dollars is invested in each of a sample of 35 income and 40 growth funds. The mean increase for a two-year period for the income funds is $900. For the growth funds, the mean increase is $875. Income funds have a sample standard deviation of $35; growth funds have a sample standard deviation of $45. Assume that the population standard deviations are equal. At the 0.05 significance level, is there a difference in the mean yields of the two funds? What decision is made about the null hypothesis using an = 0.05? 92. A company is researching the effectiveness of a new website design to decrease the time to access a website. Five website users were randomly selected and their times (in seconds) to access the website with the old and new designs were recorded. The results follow. What is the value of the test statistic to test for differences in the times? 93. A company is researching the effectiveness of a new website design to decrease the time to access a website. Five website users were randomly selected and their times (in seconds) to access the website with the old and new designs were recorded. The results follow. Let = 0.05. Is the mean time to access the new website design shorter, or is (time for the old design - time for the new design) greater than zero? Express your answer in terms of the null hypothesis. 94. Provide two examples of when a paired t-test can be used to test a hypothesis of no difference between population means. 95. For hypotheses that compare two population means, when is a pooled variance used? 96. For hypotheses that compare two population means, what test statistic is used when the population standard deviations are known? 97. For hypotheses that compare two population means, what test statistic is used when the population standard deviations are unknown? 98. When can a paired t-test be used to test a hypothesis of no difference between population means? Chapter 11 Two-Sample Tests of Hypothesis Answer Key True / False Questions 1. If the null hypothesis states that there is no difference between the mean income of males and the mean income of females, then the test is one-tailed. FALSE The test is two-tailed because we did not specify which group would have the larger mean. Also, the test is two-tailed because the null hypothesis is stated as "no difference," or H0: 1 = 2. AACSB: Communication Accessibility: Keyboard Navigation Blooms: Understand Difficulty: 1 Easy Learning Objective: 11-01 Test a hypothesis that two independent population means are equal; assuming that the population standard deviations are known and equal. Topic: Two-Sample Tests of Hypothesis: Independent Samples 2. If the null hypothesis states that there is no difference between the mean net income of retail stores in Chicago and New York City, then the test is two-tailed. TRUE AACSB: Communication Accessibility: Keyboard Navigation Blooms: Understand Difficulty: 1 Easy Learning Objective: 11-01 Test a hypothesis that two independent population means are equal; assuming that the population standard deviations are known and equal. Topic: Two-Sample Tests of Hypothesis: Independent Samples 3. When the standard deviations are equal but unknown, a test for the differences between two population means has n - 1 degrees of freedom. FALSE The degrees of freedom in the two sample test of means is found by n1 + n1 - 2. AACSB: Communication Accessibility: Keyboard Navigation Blooms: Understand Difficulty: 1 Easy Learning Objective: 11-02 Test a hypothesis that two independent population means are equal; with unknown population standard deviations. Topic: Comparing Population Means with Unknown Population Standard Deviations 4. If we are testing for the difference between two population means and assume that the two populations have equal but unknown standard deviations, the variances are pooled to compute the best estimated variance. TRUE AACSB: Communication Accessibility: Keyboard Navigation Blooms: Understand Difficulty: 1 Easy Learning Objective: 11-02 Test a hypothesis that two independent population means are equal; with unknown population standard deviations. Topic: Comparing Population Means with Unknown Population Standard Deviations 5. If we are testing for the difference between two population means and assume that the two populations have equal and unknown standard deviations, the degrees of freedom are computed as (n1)(n2) - 1. FALSE The degrees of freedom in the two sample test of means is found by n1 + n1 - 2. AACSB: Communication Accessibility: Keyboard Navigation Blooms: Understand Difficulty: 1 Easy Learning Objective: 11-02 Test a hypothesis that two independent population means are equal; with unknown population standard deviations. Topic: Comparing Population Means with Unknown Population Standard Deviations 6. A statistics professor wants to compare grades in two different classes of the same course. This is an example of a paired sample. FALSE The professor must assume the two classes are independent and use the two-sample test of means. AACSB: Communication Accessibility: Keyboard Navigation Blooms: Understand Difficulty: 1 Easy Learning Objective: 11-04 Explain the difference between dependent and independent samples. Topic: Comparing Dependent and Independent Samples 7. If we are testing for the difference between two population means, it is assumed that the sample observations from one population are independent of the sample observations from the other population. TRUE AACSB: Communication Accessibility: Keyboard Navigation Blooms: Understand Difficulty: 1 Easy Learning Objective: 11-04 Explain the difference between dependent and independent samples. Topic: Comparing Dependent and Independent Samples 8. When dependent samples are used to test for differences in the means, we compute paired differences. TRUE AACSB: Communication Accessibility: Keyboard Navigation Blooms: Understand Difficulty: 1 Easy Learning Objective: 11-04 Explain the difference between dependent and independent samples. Topic: Comparing Dependent and Independent Samples 9. If two dependent samples of size 20 are used to test the difference between the means, the degrees of freedom for a t-statistic are 19. TRUE AACSB: Communication Accessibility: Keyboard Navigation Blooms: Understand Difficulty: 1 Easy Learning Objective: 11-03 Test a hypothesis about the mean population difference between paired or dependent observations. Topic: Two-Sample Tests of Hypothesis: Dependent Samples 10. When dependent samples are used to test for differences in the means, we pool the sample variances. FALSE When dependent samples are used to test for differences in means, the sample variances are not pooled. We compute the variance of the differences between the paired observations. AACSB: Communication Accessibility: Keyboard Navigation Blooms: Understand Difficulty: 1 Easy Learning Objective: 11-03 Test a hypothesis about the mean population difference between paired or dependent observations. Topic: Two-Sample Tests of Hypothesis: Dependent Samples Multiple Choice Questions 11. A recent study focused on the number of times men and women send a Twitter message in a day. The sample information is summarized below. At the .01 significance level, is there a difference in the mean number of times men and women send a Twitter message in a day? What is the test statistic for this hypothesis? A. z-statistic B. t-statistic C. p-statistic D. df-statistic This is an independent test of means and we know the population standard deviation, hence we can use the z statistic. AACSB: Analytic Blooms: Apply Difficulty: 2 Medium Learning Objective: 11-01 Test a hypothesis that two independent population means are equal; assuming that the population standard deviations are known and equal. Topic: Two-Sample Tests of Hypothesis: Independent Samples 12. A recent study focused on the number of times men and women send a Twitter message in a day. The information is summarized next. At the .01 significance level, is there a difference in the mean number of times men and women send a Twitter message in a day? What is the value of the test statistic for this hypothesis test? A. 2.668 B. 2.672 C. 2.58 D. 2.40 To determine the value of . AACSB: Analytic Blooms: Apply Difficulty: 2 Medium Learning Objective: 11-01 Test a hypothesis that two independent population means are equal; assuming that the population standard deviations are known and equal. Topic: Two-Sample Tests of Hypothesis: Independent Samples 13. A recent study focused on the number of times men and women send a Twitter message in a day. The information is summarized next. At the .01 significance level, is there a difference in the mean number of times men and women send a Twitter message in a day? What is the p-value for this hypothesis test? A. 0.0500 B. 0.0164 C. 0.0001 D. 0.0082 To determine the value, . Because this is a two-tailed test, the p-value is the area in the tails based on the z value of 2.40. So, the p-value is . AACSB: Analytic Blooms: Apply Difficulty: 2 Medium Learning Objective: 11-01 Test a hypothesis that two independent population means are equal; assuming that the population standard deviations are known and equal. Topic: Two-Sample Tests of Hypothesis: Independent Samples 14. A recent study focused on the number of times men and women send a Twitter message in a day. The information is summarized next. At the .01 significance level, is there a difference in the mean number of times men and women send a Twitter message in a day? Based on the p-value, what is your conclusion? A. Reject the null hypothesis and conclude the means are different. B. Reject the null hypothesis and conclude the means are the same. C. Fail to reject the null hypothesis. D. Fail to reject the null hypothesis and conclude the means are different. To determine the value, . At the .01 significance level, the null hypothesis is rejected if the computed value of z is less than -2.576 or greater than 2.576. The computed value is 2.40, so we do not reject the null hypothesis. We have not demonstrated a difference in the population means. Also, based on the p-value, because this is a two-tailed test, the p-value is the area in the tails for the z value of 2.40. So, the p-value is . Therefore, we fail to reject the null hypothesis because the p-value (0.0164) is not less than the level of significance (0.01). AACSB: Analytic Blooms: Apply Difficulty: 2 Medium Learning Objective: 11-01 Test a hypothesis that two independent population means are equal; assuming that the population standard deviations are known and equal. Topic: Two-Sample Tests of Hypothesis: Independent Samples 15. A recent study focused on the amount of money single men and women save monthly. The information is summarized next. Assume that the population standard deviations are equal. At the .01 significance level, do women save more money than men? What is the test statistic for this hypothesis? A. z-statistic B. t-statistic C. p-statistic D. df-statistic We do not know the population standard deviations so the t-statistic is appropriate. AACSB: Analytic Blooms: Apply Difficulty: 2 Medium Learning Objective: 11-02 Test a hypothesis that two independent population means are equal; with unknown population standard deviations. Topic: Comparing Population Means with Unknown Population Standard Deviations 16. A recent study focused on the amount of money single men and women save monthly. The information is summarized next. Assume that the population standard deviations are equal. At the .01 significance level, do women save more money than men? What is the critical value for this hypothesis test? A. +6.213 B. +2.369 C. +2.632 D. +2.40 Applying the t-statistic and tables. There are nW + nM - 2 = 50 + 40 - 2 = 88 degrees of freedom. Move down the df column to the row with 88 degrees of freedom. Move across that row to the column headed one-tailed test and the .01 significance level. The value is +2.369. It is positive, given that the tested relationship is stated as W > M.

AACSB: Analytic

Blooms: Apply

Difficulty: 2 Medium

Learning Objective: 11-02 Test a hypothesis that two independent population means are equal; with unknown population standard deviations.

Topic: Comparing Population Means with Unknown Population Standard Deviations

17. A recent study focused on the amount of money single men and women save monthly. The information is summarized next. Assume that the population standard deviations are equal.

At the .01 significance level, do women save more money than men? What is the value of the test statistic for this hypothesis test?

A. +6.213

B. +1.318

C. +2.632

D. +2.40

The first step is to pool the sample standard deviations. . The value of the test statistic is .

AACSB: Analytic

Blooms: Apply

Difficulty: 2 Medium

Learning Objective: 11-02 Test a hypothesis that two independent population means are equal; with unknown population standard deviations.

Topic: Comparing Population Means with Unknown Population Standard Deviations

18. A recent study focused on the amount of money single men and women save monthly. The information is summarized next. Assume that the population standard deviations are equal.

At the .01 significance level, what is the conclusion about the way women and men save?

A. Reject the null hypothesis and conclude that women save more than men.

B. Reject the null hypothesis and conclude that women and men save the same amount.

C. Fail to reject the null hypothesis.

D. Fail to reject the null hypothesis and conclude the means are different.

The decision rule is to reject the null hypothesis if the computed value is greater than 2.369. To determine the value of the test statistic, the first step is to pool the sample standard deviations. . The value of the test statistic is . The computed value of the test statistic is less than the critical value, so we fail to reject the null hypothesis.

AACSB: Analytic

Blooms: Apply

Difficulty: 2 Medium

Learning Objective: 11-02 Test a hypothesis that two independent population means are equal; with unknown population standard deviations.

Topic: Comparing Population Means with Unknown Population Standard Deviations

19. If the null hypothesis that two means are equal is true, where will 97% of the computed z values lie between?

A. 2.58

B. 2.33

C. 2.17

D. 2.07

This is a two-tailed test, so first we split the significance level in half. The significance level is 1.00 0.97 = 0.03. So 0.03/2 = .015 of the area will be in each tail of the z distribution. To find the z values, we subtract the area in the tail from .5000, so .5000 .0150 = .485. Next, we search the body of the areas under the normal curve table for a value as close to .4850 as possible. We locate this value and find the values 2.1 in the margin of the row and .07 in the margin of the column. So 97% of the values are between -2.17 and 2.17.

AACSB: Analytic

Accessibility: Keyboard Navigation

Blooms: Apply

Difficulty: 3 Hard

Learning Objective: 11-01 Test a hypothesis that two independent population means are equal; assuming that the population standard deviations are known and equal.

Topic: Two-Sample Tests of Hypothesis: Independent Samples

20. Assuming the population variances are known, the population variance of the difference between two means is _____________.

A. The sum of the two means

B. The sum of the two population variances

C. The sum of the two population standard deviations

D. The sum of the two sample sizes for each population

We assume the means are from independent populations; thus, the variance of the difference is the sum of the variances .

AACSB: Communication

Blooms: Understand

Difficulty: 3 Hard

Learning Objective: 11-01 Test a hypothesis that two independent population means are equal; assuming that the population standard deviations are known and equal.

Topic: Two-Sample Tests of Hypothesis: Independent Samples

21. When testing the difference between two population means, the sample variances are pooled to estimate the population variance when ________________.

A. The population variances are known and equal

B. The population means are known

C. The population variances are assumed unequal and unknown

D. The population variances are assumed equal but unknown

We pool the sample variances when the population variances are not known and we assume the populations have the same variances. The pooled sample variances are the best point estimate of the population variance.

AACSB: Communication

Accessibility: Keyboard Navigation

Blooms: Understand

Difficulty: 3 Hard

Learning Objective: 11-02 Test a hypothesis that two independent population means are equal; with unknown population standard deviations.

Topic: Comparing Population Means with Unknown Population Standard Deviations

22. When testing the difference between two dependent population means, the test statistic is based on a ______________.

A. Pooled variance

B. Standard deviation of the differences

C. Pooled proportion

D. Sum of the population variances

When the samples are dependent, we find the difference between the paired or related observations. The test statistic is based on the standard deviation of these differences.

AACSB: Communication

Accessibility: Keyboard Navigation

Blooms: Understand

Difficulty: 3 Hard

Learning Objective: 11-03 Test a hypothesis about the mean population difference between paired or dependent observations.

Topic: Two-Sample Tests of Hypothesis: Dependent Samples

23. The net weights (in grams) of a sample of bottles filled by a machine manufactured by Edne, and the net weights of a sample filled by a similar machine manufactured by Orno, Inc., are:

Testing the claim at the 0.05 level that the mean weight of the bottles filled by the Orno machine is greater than the mean weight of the bottles filled by the Edne machine, what is the critical value? Assume equal standard deviations for both samples.

A. +2.179

B. +2.145

C. +1.782

D. +1.761

For the hypotheses: H0: O E; H1: O > E, this test will be based on a t statistic because it is all sample data. There are degrees of freedom. We wish to show the mean weight filled by Orno is larger, hence this is a one (right)-tailed test. Go to t table, move down the left column to 12 degrees of freedom, then move to the column headed .05 and a one-tailed test. The value is +1.782.

AACSB: Analytic

Blooms: Apply

Difficulty: 3 Hard

Learning Objective: 11-02 Test a hypothesis that two independent population means are equal; with unknown population standard deviations.

Topic: Comparing Population Means with Unknown Population Standard Deviations

24. Which condition must be met to conduct a test for the difference in two sample means using a z-statistic?

A. The data must be at least of nominal scale.

B. The populations must be normal.

C. The two population standard deviations must be known.

D. The samples are dependent.

To test for a difference between two means with a z statistic, it is necessary that we know the population standard deviations.

AACSB: Communication

Accessibility: Keyboard Navigation

Blooms: Understand

Difficulty: 2 Medium

Learning Objective: 11-01 Test a hypothesis that two independent population means are equal; assuming that the population standard deviations are known and equal.

Topic: Two-Sample Tests of Hypothesis: Independent Samples

25. When is it appropriate to use the paired difference t-test?

A. When four samples are compared at once

B. When any two samples are compared

C. When two independent samples are compared

D. When two dependent samples are compared

To use the paired difference t-test, the two samples must be dependent or related.

AACSB: Communication

Accessibility: Keyboard Navigation

Blooms: Understand

Difficulty: 2 Medium

Learning Objective: 11-04 Explain the difference between dependent and independent samples.

Topic: Comparing Dependent and Independent Samples

26. We test for a hypothesized difference between two population means: H0: 1 = 2. The population standard deviations are unknown but assumed equal. The number of observations in the first sample is 15, and 12 in the second sample. How many degrees of freedom are associated with the critical value?

A. 24

B. 25

C. 26

D. 27

The test statistic is a t statistic, and the associated degrees of freedom are found by calculating .

AACSB: Communication

Blooms: Understand

Difficulty: 2 Medium

Learning Objective: 11-02 Test a hypothesis that two independent population means are equal; with unknown population standard deviations.

Topic: Comparing Population Means with Unknown Population Standard Deviations

27. For a hypothesis comparing two population means, H0: 1 2, what is the critical value for a one-tailed hypothesis test, using a 5% significance level, with both sample sizes equal to 13? Assume the population standard deviations are equal.

A. 1.711

B. +1.711

C. +2.060

D. +2.064

The test statistic is a t statistic and the associated degrees of freedom are . This is a one-tailed test, so go to the t table, move down the left column to 24 degrees of freedom, then move to the column headed .05 and a one-tailed test. The value is +1.711. The sign is positive because the alternate hypothesis is H1: 1 > 2.

Blooms: Apply

Difficulty: 2 Medium

Learning Objective: 11-02 Test a hypothesis that two independent population means are equal; with unknown population standard deviations.

Topic: Comparing Population Means with Unknown Population Standard Deviations

28. For a hypothesis test comparing two population means, the combined degrees of freedom are 24. Which of the following statements about the two sample sizes is NOT true? Assume the population standard deviations are equal.

A. n1 = 11; n2 = 13

B. n1 = 12; n2 = 14

C. n1 = 13; n2 = 13

D. n1 = 10; n2 = 16

The total number of degrees of freedom is 24, so . From this equation, the sum of the two samples must be 26. Only the answer has sample sizes that do not sum to 26.

AACSB: Reflective Thinking

Blooms: Analyze

Difficulty: 3 Hard

Learning Objective: 11-02 Test a hypothesis that two independent population means are equal; with unknown population standard deviations.

Topic: Comparing Population Means with Unknown Population Standard Deviations

29. Two samples, one of size 14 and the second of size 13, are selected to test the difference between two population means. How many degrees of freedom are used to find the critical value? Assume the population standard deviations are equal.

A. 27

B. 26

C. 25

D. 14

The hypothesis test will be based on the t statistic. There are 25 degrees of freedom: .

AACSB: Communication

Blooms: Understand

Difficulty: 3 Hard

Learning Objective: 11-02 Test a hypothesis that two independent population means are equal; with unknown population standard deviations.

Topic: Comparing Population Means with Unknown Population Standard Deviations

30. Twenty randomly selected statistics students were given 15 multiple-choice questions and 15 open-ended questions, all on the same material. The professor was interested in determining if students scored higher on the multiple-choice questions. This experiment is an example of ________________.

A. A one-sample test of means

B. A two-sample test of means

C. A paired t-test

D. A test of proportions

For each student, we have two scores, number of correct multiple-choice questions and number of correct open-ended questions. The samples are dependent. For each student, we can compute the difference between the number correct between the two types of questions and test H0: d 0.

AACSB: Reflective Thinking

Accessibility: Keyboard Navigation

Blooms: Analyze

Difficulty: 2 Medium

Learning Objective: 11-03 Test a hypothesis about the mean population difference between paired or dependent observations.

Topic: Two-Sample Tests of Hypothesis: Dependent Samples

31. A national manufacturer of ball bearings is experimenting with two different processes for producing precision ball bearings. It is important that the diameters be as close as possible to an industry standard. The output from each process is sampled and the average error from the industry standard is measured in millimeters. The results are presented next.

The researcher is interested in determining whether there is evidence that the two processes yield different average errors. The population standard deviations are unknown but are assumed equal. What is the null hypothesis?

A. H0: A = B

B. H0: A B

C. H0: A B

D. H0: A > B

The two populations are independent and we wish to investigate whether there is a difference in the means. So the null hypothesis is H0: A = B.

AACSB: Communication

Blooms: Understand

Difficulty: 2 Medium

Learning Objective: 11-02 Test a hypothesis that two independent population means are equal; with unknown population standard deviations.

Topic: Comparing Population Means with Unknown Population Standard Deviations

32. A national manufacturer of ball bearings is experimenting with two different processes for producing precision ball bearings. It is important that the diameters be as close as possible to an industry standard. The output from each process is sampled and the average error from the industry standard is measured in millimeters. The results are presented next.

The researcher is interested in determining whether there is evidence that the two processes yield different average errors. The population standard deviations are unknown but assumed equal. What is the alternate hypothesis?

A. H1: A = B

B. H1: A B

C. H1: A B

D. H1: A > B

The two populations are independent and we wish to investigate whether there is a difference in the means. So the null hypothesis is H0: A = B and the alternate hypothesis is that the means are not the same or H1: A B.

AACSB: Communication

Blooms: Understand

Difficulty: 2 Medium

Learning Objective: 11-02 Test a hypothesis that two independent population means are equal; with unknown population standard deviations.

Topic: Comparing Population Means with Unknown Population Standard Deviations

33. A national manufacturer of ball bearings is experimenting with two different processes for producing precision ball bearings. It is important that the diameters be as close as possible to an industry standard. The output from each process is sampled and the average error from the industry standard is measured in millimeters. The results are presented next.

The researcher is interested in determining whether there is evidence that the two processes yield different average errors. The population standard deviations are unknown but are assumed equal. What are the degrees of freedom?

A. 10

B. 13

C. 26

D. 24

The hypothesis test is based on a t-statistic with the associated degrees of freedom: .

Blooms: Understand

Difficulty: 2 Medium

Learning Objective: 11-02 Test a hypothesis that two independent population means are equal; with unknown population standard deviations.

Topic: Comparing Population Means with Unknown Population Standard Deviations

34. A national manufacturer of ball bearings is experimenting with two different processes for producing precision ball bearings. It is important that the diameters be as close as possible to an industry standard. The output from each process is sampled and the average error from the industry standard is measured in millimeters. The results are presented next.

The researcher is interested in determining whether there is evidence that the two processes yield different average errors. The population standard deviations are unknown but assumed equal. What is the critical t value at the 1% level of significance?

A. +2.797

B. -2.492

C. 1.711

D. 2.797

The hypothesis test, H0: A = B, is based on a t-statistic with degrees of freedom: . Go to the t table, go down the left-hand column to 24 degrees of freedom, move across that row to the column labeled: two-tailed test at the .01 significance level. The value is 2.797 because it is a two-tailed hypothesis test.

Blooms: Apply

Difficulty: 2 Medium

Learning Objective: 11-02 Test a hypothesis that two independent population means are equal; with unknown population standard deviations.

Topic: Comparing Population Means with Unknown Population Standard Deviations

35. A national manufacturer of ball bearings is experimenting with two different processes for producing precision ball bearings. It is important that the diameters be as close as possible to an industry standard. The output from each process is sampled and the average error from the industry standard is measured in millimeters. The results are presented next.

The researcher is interested in determining whether there is evidence that the two processes yield different average errors. The population standard deviations are unknown but are assumed equal. What is the computed value of t?

A. +2.797

B. -1.000

C. -3.299

D. 0.5938

The first step is to pool the standard deviations:

The second step is

AACSB: Analytic

Blooms: Apply

Difficulty: 3 Hard

Learning Objective: 11-02 Test a hypothesis that two independent population means are equal; with unknown population standard deviations.

Topic: Comparing Population Means with Unknown Population Standard Deviations

36. A national manufacturer of ball bearings is experimenting with two different processes for producing precision ball bearings. It is important that the diameters be as close as possible to an industry standard. The output from each process is sampled and the average error from the industry standard is measured in millimeters. The results are presented next.

The researcher is interested in determining whether there is e

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