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.

SCENARIO 10-16-B A survey on men and women's shopping behavior collected data from a sample of 50 men and 50 women. The following data on online shopping spending every month are stored in the file Scenario10-16-DataB.XLSX. Men 122 184 220 61 114 293 107 200 242 233 229 157 203 147 110 217 136 222 206 155 292 220 210 196 177 212 169 243 240 78 226 173 152 228 230 162 189 171 184 263 217 170 191 176 129 241 208 217 251 267 Women 170 180 167 232 217 145 190 145 200 163 159 138 193 170 165 172 190 127 186 245 198 186 163 129 158 164 197 182 172 163 180 168 158 185 166 216 153 146 197 129 162 202 192 119 197 172 139 167 203 130 Treat the men as population 1 and the women as population 2. -Referring to Scenario 10-16-B, what is(are) the critical value(s) when testing whether there is evidence of a difference in the means of monthly online shopping spending between men and women if the level of significance is 0.05 based on the conclusion that you have reached in testing for evidence of a difference in the variances?

The Wall Street Journal recently ran an article indicating differences in perception of sexual harassment on the job between men and women. The article claimed that women perceived the problem to be much more prevalent than did men. One question asked to both men and women was: "Do you think sexual harassment is a major problem in the American workplace?" Some 24% of the men compared to 62% of the women responded "Yes." Suppose that 150 women and 200 men were interviewed. Construct a 95% confidence interval estimate of the difference between the proportion of women and men who think sexual harassment is a major problem in the American workplace.

SCENARIO 10-13 The amount of time required to reach a customer service representative has a huge impact on customer satisfaction. Below is the Excel output from a study to see whether there is evidence of a difference in the mean amounts of time required to reach a customer service representative between two hotels. Assume that the population variances in the amount of time for the two hotels are not equal. t-Test: Two-Sample Assuming Unequal Variances Hotel 1 Hotel 2 Mean 2.214 2.0115 Variance 2.951657 3.57855 Observations 20 20 Hypothesized Mean Difference 0 df 38 t Stat 0.354386 P(T<=t) one-tail 0.362504 Critical one-tail 1.685953 (<=) two-tail 0.725009 Critical two-tail 2.024394 -Referring to Scenario 10-13, what is the value of the test statistic for testing if there is evidence of a difference in the variabilities of the amount of time required to reach a customer service representative between the two hotels?

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 90% confidence interval estimate of the difference between the proportion of men and women who believe that sexual discrimination is a problem.

SCENARIO 10-11 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 Scenario 10-11, the p-value of the test is ________.

SCENARIO 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 Scenario 10-5, the calculated value of the test statistic is ________.

SCENARIO 10-6 To investigate the efficacy of a diet, a random sample of 16 male patients is selected from a population of adult males using the diet. The weight of each individual in the sample is taken at the start of the diet and at a medical follow-up 4 weeks later. Assuming that the population of differences in weight before versus after the diet follow a normal distribution, the t-test for related samples can be used to determine if there was a significant decrease in the mean weight during this period. Suppose the mean decrease in weights over all 16 subjects in the study is 3.0 pounds with the standard deviation of differences computed as 6.0 pounds. -Referring to Scenario 10-6, what is the 99% confidence interval estimate for the mean difference in weight before and after the diet?

SCENARIO 10-7 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. -Referring to Scenario 10-7, if the buyer had decided to perform a two-tail test, the p-value would have been ________.

SCENARIO 10-10 A corporation randomly selects 150 salespeople and finds that 66% who have never taken a self- improvement course would like such a course. The firm did a similar study 10 years ago in which 60% of a random sample of 160 salespeople wanted a self-improvement course. The groups are assumed to be independent random samples. Let $\pi _ { 1 } \text { and } \pi _ { 2 }$ represent the true proportion of workers who would like to attend a self-improvement course in the recent study and the past study, respectively. -Referring to Scenario 10-10, what is the estimated standard error of the difference between the two sample proportions?

SCENARIO 10-12 A quality control engineer is in charge of the manufacture of USB flash drives. Two different processes can be used to manufacture the flash drives. He suspects that the Kohler method produces a greater proportion of defects than the Russell method. He samples 150 of the Kohler and 200 of the Russell flash drives and finds that 27 and 18 of them, respectively, are defective. If Kohler is designated as "Group 1" and Russell is designated as "Group 2," perform the appropriate test at a level of significance of 0.01. -Referring to Scenario 10-12, the hypotheses that should be tested are: a) $H _ { 0 } : \pi _ { 1 } - \pi _ { 2 } = 0$ versus $H _ { 1 } : \pi _ { 1 } - \pi _ { 2 } \neq 0$ b) $H _ { 0 } : \pi _ { 1 } - \pi _ { 2 } \neq 0$ versus $H _ { 1 } : \pi _ { 1 } - \pi _ { 2 } = 0$ c) $H _ { 0 } : \pi _ { 1 } - \pi _ { 2 } \leq 0$ versus $H _ { 1 } : \pi _ { 1 } - \pi _ { 2 } > 0$ d) $H _ { 0 } : \pi _ { 1 } - \pi _ { 2 } \geq 0$ versus $H _ { 1 } : \pi _ { 1 } - \pi _ { 2 } < 0$

SCENARIO 10-13 The amount of time required to reach a customer service representative has a huge impact on customer satisfaction. Below is the Excel output from a study to see whether there is evidence of a difference in the mean amounts of time required to reach a customer service representative between two hotels. Assume that the population variances in the amount of time for the two hotels are not equal. t-Test: Two-Sample Assuming Unequal Variances Hotel 1 Hotel 2 Mean 2.214 2.0115 Variance 2.951657 3.57855 Observations 20 20 Hypothesized Mean Difference 0 df 38 t Stat 0.354386 P(T<=t) one-tail 0.362504 Critical one-tail 1.685953 (<=) two-tail 0.725009 Critical two-tail 2.024394 -Referring to Scenario 10-13, what is the smallest level of significance at which the null hypothesis will still not be rejected?

SCENARIO 10-8 A few years ago, Pepsi invited consumers to take the "Pepsi Challenge." Consumers were asked to decide which of two sodas, Coke or Pepsi, they preferred in a blind taste test. Pepsi was interested in determining what factors played a role in people's taste preferences. One of the factors studied was the gender of the consumer. Below are the results of analyses comparing the taste preferences of men and women with the proportions depicting preference for Pepsi. Males: n=109,=0.422018 Females: n=52,=0.25 -=0.172018Z=2.11825 -Referring to Scenario 10-8, to determine if a difference exists in the taste preferences of men and women, give the correct alternative hypothesis that Pepsi would test. a) $H _ { 1 } : \mu _ { M } - \mu _ { F } \neq 0$ b) $H _ { 1 } : \mu _ { M } - \mu _ { F } > 0$ c) $H _ { 1 } : \pi _ { M } - \pi _ { F } \neq 0$ d) $H _ { 1 } : \pi _ { M } - \pi _ { F } = 0$

The Wall Street Journal recently ran an article indicating differences in perception of sexual harassment on the job between men and women. The article claimed that women perceived the Problem to be much more prevalent than did men. One question asked to both men and women Was: "Do you think sexual harassment is a major problem in the American workplace?" Some 24% of the men compared to 62% of the women responded "Yes." Suppose that 150 women and 200 men were interviewed. What is the value of the test statistic?

SCENARIO 10-7 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. -Referring to Scenario 10-7, the buyer should decide that the primary supplier is

SCENARIO 10-3 A real estate company is interested in testing whether the mean time that families in Gotham have been living in their current homes is less than families in Metropolis. 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: $\bar { X } _ { \mathrm { G } } = 35 \text { months, } \quad S _ { \mathrm { G } } { } ^ { 2 } = 900 \quad \text { Metropolis: } \quad \bar { X } _ { \mathrm { M } } = 50 \text { months, } \quad S _ { \mathrm { M } } ^ { 2 } = 1050$ -Referring to Scenario 10-3, which of the following represents the relevant hypotheses tested by the real estate company? a) $H _ { 0 } : \mu _ { G } - \mu _ { M } \geq 0$ versus $H _ { 1 } : \mu _ { G } - \mu _ { M } < 0$ b) $H _ { 0 } : \mu _ { G } - \mu _ { M } \leq 0$ versus $H _ { 1 } : \mu _ { G } - \mu _ { M } > 0$ c) $H _ { 0 } : \underline { \mu } _ { G } - \underline { \mu } _ { M } = 0$ versus $H _ { 1 } : \underline { \mu } _ { G } - \underline { \mu } _ { M } \neq 0$ d) $H _ { 0 } : \bar { X } _ { G } - \bar { X } _ { M } \geq 0$ versus $H _ { 1 } : \bar { X } _ { G } - \bar { X } _ { M } < 0$

SCENARIO 10-15 The table below presents the summary statistics for the starting annual salaries (in thousands of dollars) for individuals entering the public accounting and financial planning professions. Sample I (public accounting): $\bar { X } _ { 1 } = 60.35 , S _ { 1 } = 3.25 , n _ { 1 } = 12$ Sample II (financial planning): $\bar { X } _ { 2 } = 58.20 , S _ { 2 } = 2.48 , n _ { 2 } = 14$ Test whether the mean starting annual salaries for individuals entering the public accounting professions is higher than that of financial planning assuming that the two population variances are the same. -Referring to Scenario 10-15, which of the following represents the relevant hypotheses tested? a) $H _ { 0 } : \mu _ { 1 } - \mu _ { 2 } \geq 0$ versus $H _ { 1 } : \mu _ { 1 } - \mu _ { 2 } < 0$ b) $H _ { 0 } : \mu _ { 1 } - \mu _ { 2 } \leq 0$ versus $H _ { 1 } : \mu _ { 1 } - \mu _ { 2 } > 0$ c) $H _ { 0 } : \mu _ { 1 } - \mu _ { 2 } = 0$ versus $H _ { 1 } : \mu _ { 1 } - \mu _ { 2 } \neq 0$ d) $H _ { 0 } : \mu _ { 1 } - \mu _ { 2 } \neq 0$ versus $H _ { 1 } : \mu _ { 1 } - \mu _ { 2 } = 0$

SCENARIO 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 Scenario 10-5, the number of degrees of freedom is

SCENARIO 10-3 A real estate company is interested in testing whether the mean time that families in Gotham have been living in their current homes is less than families in Metropolis. 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: $\bar { X } _ { \mathrm { G } } = 35 \text { months, } \quad S _ { \mathrm { G } } { } ^ { 2 } = 900 \quad \text { Metropolis: } \quad \bar { X } _ { \mathrm { M } } = 50 \text { months, } \quad S _ { \mathrm { M } } ^ { 2 } = 1050$ -Referring to Scenario 10-3, what is the test statistic for the difference between sample means?

SCENARIO 10-4 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 Scenario 10-4, a two-tail test of the null hypothesis of no difference would _______ (be rejected/not be rejected) at the $\alpha = 0.05$ level of significance.