Exam 10: Two-Sample Tests

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 null hypothesis will be rejected if the test statistic is ________.

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, suppose that the two-tail p-value was really 0.0734. State the proper conclusion. a) At $\alpha = 0.05$ , there is sufficient evidence to indicate the proportion of males preferring Pepsi differs from the proportion of females preferring Pepsi. b) At $\alpha = 0.10$ , there is sufficient evidence to indicate the proportion of males preferring Pepsi differs from the proportion of females preferring Pepsi. c) At $\alpha = 0.05$ , there is sufficient evidence to indicate the proportion of males preferring Pepsi equals the proportion of females preferring Pepsi. d) At $\alpha = 0.08$ , there is insufficient evidence to indicate the proportion of males preferring Pepsi differs from the proportion of females preferring Pepsi.

SCENARIO 10-14 The use of preservatives by food processors has become a controversial issue. Suppose two preservatives are extensively tested and determined safe for use in meats. A processor wants to compare the preservatives for their effects on retarding spoilage. Suppose 15 cuts of fresh meat are treated with preservative I and 15 are treated with preservative II, and the number of hours until spoilage begins is recorded for each of the 30 cuts of meat. The results are summarized in the table below. Preservative I Preservative II Preservative I Preservative II =106.4 hours =96.54 hours =10.3 hours =13.4 hours -Referring to Scenario 10-14, what assumptions are necessary for testing if the population variances differ for preservatives I and II?

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 99% confidence interval estimate of the difference between the proportion of men and women who believe that sexual discrimination is a problem.

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. For a 0.01 level of significance, what is the critical value for the Rejection region?

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 10-12, the hypotheses the dean should use 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 10-3 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(are) the critical value(s) of the relevant hypothesis test if the level of significance is 0.05? a) $t \cong Z = - 1.645$ b) $t \cong Z = \pm 1.96$ c) $t \cong Z = - 1.96$ d) $t \cong Z = - 2.080$

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, 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? 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 10-16-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 the value of the test statistic when testing whether there is evidence of a difference in the variances of monthly online shopping spending between men and women?

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, suppose ? = 0.10. Which of the following represents the correct conclusion?

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 same decision would be made with this test if the level of significance had been 0.10 rather than 0.05.

SCENARIO 10-16-A 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-DataA.XLSX. Men 148 211 256 309 190 205 203 208 231 125 149 205 195 178 196 198 110 199 181 168 218 222 206 168 239 130 246 149 262 142 174 181 198 147 143 185 200 166 171 133 Women 295 242 299 209 189 173 109 291 208 227 272 176 251 235 145 179 30 135 200 270 174 123 199 195 192 102 110 184 228 316 170 234 163 245 174 126 227 86 96 185 288 154 217 184 154 217 222 175 265 196 172 113 240 235 269 243 131 134 56 164 Treat the men as population 1 and the women as population 2. -Referring to Scenario 10-16-A, the normality probability plot suggests that monthly online shopping spending for the men is quite normally distributed.

SCENARIO 10-16-A 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-DataA.XLSX. Men 148 211 256 309 190 205 203 208 231 125 149 205 195 178 196 198 110 199 181 168 218 222 206 168 239 130 246 149 262 142 174 181 198 147 143 185 200 166 171 133 Women 295 242 299 209 189 173 109 291 208 227 272 176 251 235 145 179 30 135 200 270 174 123 199 195 192 102 110 184 228 316 170 234 163 245 174 126 227 86 96 185 288 154 217 184 154 217 222 175 265 196 172 113 240 235 269 243 131 134 56 164 Treat the men as population 1 and the women as population 2. -Referring to Scenario 10-16-A, what is the highest level of significance at which a test on a difference in the variances will not be rejected?

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, the critical values for a two-tail test of the null hypothesis of no difference in the population means at the $\alpha = 0.05$ level of significance are _______.

SCENARIO 10-16-A 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-DataA.XLSX. Men 148 211 256 309 190 205 203 208 231 125 149 205 195 178 196 198 110 199 181 168 218 222 206 168 239 130 246 149 262 142 174 181 198 147 143 185 200 166 171 133 Women 295 242 299 209 189 173 109 291 208 227 272 176 251 235 145 179 30 135 200 270 174 123 199 195 192 102 110 184 228 316 170 234 163 245 174 126 227 86 96 185 288 154 217 184 154 217 222 175 265 196 172 113 240 235 269 243 131 134 56 164 Treat the men as population 1 and the women as population 2. -Referring to Scenario 10-16-A, the normality probability plot suggests that monthly online shopping spending for the women is quite normally distributed.

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, the normality probability plot suggests that monthly online shopping spending for the women is quite normally distributed.

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, what is(are) the critical value(s) of the relevant hypothesis test if the level of significance is 0.10?

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, the critical value for a one-tail test of the null hypothesis of no difference at the $\alpha = 0.05$ level of significance is _______.

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, the company tests to determine at the 0.05 level whether the population proportion has changed from the previous study. Which of the following is correct?

SCENARIO 10-14 The use of preservatives by food processors has become a controversial issue. Suppose two preservatives are extensively tested and determined safe for use in meats. A processor wants to compare the preservatives for their effects on retarding spoilage. Suppose 15 cuts of fresh meat are treated with preservative I and 15 are treated with preservative II, and the number of hours until spoilage begins is recorded for each of the 30 cuts of meat. The results are summarized in the table below. Preservative I Preservative II Preservative I Preservative II =106.4 hours =96.54 hours =10.3 hours =13.4 hours -Referring to Scenario 10-14, state the null and alternative hypotheses for testing if the population variances differ for preservatives I and II. a) $H _ { 0 } : \sigma _ { I } ^ { 2 } - \sigma _ { I I } ^ { 2 } \geq 0$ versus $H _ { 1 } : \sigma _ { I } ^ { 2 } - \sigma _ { I I } ^ { 2 } < 0$ b) $H _ { 0 } : \sigma _ { I } ^ { 2 } - \sigma _ { I I } ^ { 2 } \leq 0$ versus $H _ { 1 } : \sigma _ { I } ^ { 2 } - \sigma _ { I I } ^ { 2 } > 0$ c) $H _ { 0 } : \sigma _ { I } ^ { 2 } - \sigma _ { I I } ^ { 2 } = 0$ versus $H _ { 1 } : \sigma _ { I } ^ { 2 } - \sigma _ { I I } ^ { 2 } \neq 0$ d) $H _ { 0 } : \sigma _ { I } ^ { 2 } - \sigma _ { I I } ^ { 2 } \neq 0$ versus $H _ { 1 } : \sigma _ { I } ^ { 2 } - \sigma _ { I I } ^ { 2 } = 0$