Exam 10: Two-Sample Tests

A powerful women's group has claimed that men and women differ in attitudes about sexual discrimination. A group of 50 men (group 1) and 40 women (group 2) were asked if they thought sexual discrimination is a problem in the United States. Of those sampled, 11 of the men and 19 of the women did believe that sexual discrimination is a problem. Construct a 95% confidence interval estimate of the difference between the proportion of men and women who believe that sexual discrimination is a problem.

A researcher is curious about the effect of sleep on students' test performances. He chooses 60 students and gives each 2 tests: one given after 2 hours' sleep and one after 8 hours' sleep. The test the researcher should use would be a related samples test.

TABLE 10-5 To test the effectiveness of a business school preparation course, 8 students took a general business test before and after thecourse. The results are given below Exam Score Exam Score Student Before Course (1) After Course (2) 1 530 670 2 690 770 3 910 1,000 4 700 710 5 450 550 6 820 870 7 820 770 8 630 610 -Referring to Table 10-11, if the firm wanted to test whether a greater proportion of workers would currently like to attend a self-improvement course than in the past, which represents the relevant hypotheses?

TABLE 10-5 To test the effectiveness of a business school preparation course, 8 students took a general business test before and after the course. The results are given below. Exam Score Exam Score Student Before Course (1) After Course (2) 1 530 670 2 690 770 3 910 1,000 4 700 710 5 450 550 6 820 870 7 820 770 8 630 610 -Referring to Table 10-5, at the 0.05 level of significance, the conclusion for this hypothesis test would be

TABLE 10-4 A real estate company is interested in testing whether, on average, families in Gotham have been living in their current homes for less time than families in Metropolis have. Assume that the two population variances are equal. A random sample of 100 families from Gotham and a random sample of 150 families in Metropolis yield the following data on length of residence in current homes. Gotham: $\overline{X}$ _G = 35 months, s_G² = 900 $\quad$$\quad$ Metropolis: $\overline{X}$ _M = 50 months, s_M² = 1050 -Referring to Table 10-4, suppose ? = 0.01. Which of the following represents the correct conclusion?

$\text { TABLE 10-5 }$ To test the effectiveness of a business school preparation course, 8 students took a general business test before and after the course. The results are given below. Student Exam Score Before Course (1) Exam Score After Course (2) 1 530 670 2 690 770 3 910 1,000 4 700 710 5 450 550 6 820 870 7 820 770 8 630 610 -Referring to Table 10-5, the value of the sample mean difference is ______if the difference scores reflect the results of the exam after the course minus the results of the exam before the course.

In testing for the differences between the means of two related populations, we assume that the differences follow a(n) _____distribution.

If we wish to determine whether there is evidence that the proportion of items of interest is higher in Group 1 than in Group 2, and the test statistic for Z = +2.07 where the difference is defined as Group 1's proportion minus Group 2's proportion, the p-value is equal to______ .

TABLE 10-14 A problem with a telephone line that prevents a customer from receiving or making calls is disconcerting to both the customer and the telephone company. The data on samples of 20 problems reported to two different offices of a telephone company and the time to clear these problems (in minutes) from the customers' lines are collected. Below is the Excel output to see whether there is evidence of a difference in the mean waiting time between the two offices assuming that the population variances in the two offices are not equal. t- Test: Two- Sample Assuming Unequal Variances Office 1 Office 2 Mean 2214 2.0115 Variance 2.951657 3.57855 Observations 20 20 Hypothesized Mean Difference 0 38 Stat 0.354386 (<=) one- tail 0.362504 Critical one- tail 1.685953 (<=) two- tail 0.725009 Critical two- tail 2.024394 -Referring to Table 10-14, what is the upper critical value for testing if there is evidence of a difference in the variability of the waiting time between the two offices at the 5% level of significance?

The F distribution can only have positive values.

For all two-sample tests, the sample sizes must be equal in the two groups.

Given the following information, calculate the degrees of freedom that should be used in the pooled-variance t test. =4 =6 =16 =25

If we are testing for the difference between the means of two independent populations presuming equal variances with samples of n1 = 20 and n2= 20, the number of degrees of freedom is equal to

TABLE 10-12 The dean of a college is interested in the proportion of graduates from his college who have a job offer on graduation day. He is particularly interested in seeing if there is a difference in this proportion for accounting and economics majors. In a random sample of 100 of each type of major at graduation, he found that 65 accounting majors and 52 economics majors had job offers. If the accounting majors are designated as "Group 1" and the economics majors are designated as "Group 2," perform the appropriate hypothesis test using a level of significance of 0.05. -Referring to Table 10-12, the same decision would be made with this test if the level of significance had been 0.01 rather than 0.05.

TABLE 10-12 The dean of a college is interested in the proportion of graduates from his college who have a job offer on graduation day. He is particularly interested in seeing if there is a difference in this proportion for accounting and economics majors. In a random sample of 100 of each type of major at graduation, he found that 65 accounting majors and 52 economics majors had job offers. If the accounting majors are designated as "Group 1" and the economics majors are designated as "Group 2," perform the appropriate hypothesis test using a level of significance of 0.05. -Referring to Table 10-12, the hypotheses the dean should use are A) $H_{0}: \pi_{1}-\pi_{2} \leq 0$ versus $H_{1}: \pi_{1}-\pi_{2}>0$ . B) $H_{0}: \pi_{1}-\pi_{2}=0$ versus $H_{1}: \pi_{1}-\pi_{2} \neq 0$ . C) $H_{0}: \pi_{1}-\pi_{2} \geq 0$ versus $H_{1}: \pi_{1}-\pi_{2}<0$ . D) $H_{0}: \pi_{1}-\pi_{2} \neq 0$ versus $H_{1}: \pi_{1}-\pi_{2}=0$ .

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

TABLE 10-8 A buyer for a manufacturing plant suspects that his primary supplier of raw materials is overcharging. In order to determine if his suspicion is correct, he contacts a second supplier and asks for the prices on various identical materials. He wants to compare these prices with those of his primary supplier. The data collected is presented in the table below, with some summary statistics presented (all of these might not be necessary to answer the questions which follow). The buyer believes that the differences are normally distributed and will use this sample to perform an appropriate test at a level of significance of 0.01. Primary Secondary 1 \ 55 \ 45 \ 10 2 \ 48 \ 47 \ 1 3 \ 31 \ 32 -\ 1 4 \ 83 \ 77 \ 6 5 \ 37 \ 37 \ 0 6 \ 55 \ 54 \ 1 Sum: \ 309 \ 292 \ 17 Sum of Squares: \ 17,573 \ 15,472 \ 139 -Referring to Table 10-8, the calculated value of the test statistic is_____ .

TABLE 10-6 Two samples each of size 25 are taken from independent populations assumed to be normally distributed with equal variances. The first sample has a mean of 35.5 and standard deviation of 3.0 while the second sample has a mean of 33.0 and standard deviation of 4.0. -Referring to Table 10-6, what is the 90% confidence interval estimate for the difference in the two means?

TABLE 10-4 A real estate company is interested in testing whether, on average, families in Gotham have been living in their current homes for less time than families in Metropolis have. Assume that the two population variances are equal. A random sample of 100 families from Gotham and a random sample of 150 families in Metropolis yield the following data on length of residence in current homes. $\text { Gotham: } \bar{X} _ { G } = 35 \text { months, } s _G ^ { 2 } = 900 \text { Metropolis: } \bar{X} _ { M } = 50 \text { months, } s _ { M ^ { 2 } } = 1050$ -Referring to Table 10-4, what is the standardized value of the estimate of the mean of the sampling distribution of the difference between sample means?

TABLE 10-14 A problem with a telephone line that prevents a customer from receiving or making calls is disconcerting to both the customer and the telephone company. The data on samples of 20 problems reported to two different offices of a telephone company and the time to clear these problems (in minutes) from the customers' lines are collected. Below is the Excel output to see whether there is evidence of a difference in the mean waiting time between the two offices assuming that the population variances in the two offices are not equal. $\text { t-Test: Two- Sample Assuming Unequal Variances }$ Office 1 Office 2 Mean 2214 2.0115 Variance 2.951657 3.57855 Observations 20 20 Hypothesized Mean Difference 0 38 Stat 0.354386 (<=) one- tail 0.362504 Critical one- tail 1.685953 (<=) two- tail 0.725009 Critical two-tail 2.024394 -Referring to Table 10-14, what is the value of the test statistic for testing if there is evidence of a difference in the variability of the waiting time between the two offices?