Exam 10: Two-Sample Tests

SCENARIO 10-4 An agronomist wants to compare the crop yield of 3 varieties of chickpea seeds.She plants 15 fields, 5 with each variety.She then measures the crop yield in bushels per acre.Treating this as a completely randomized design, the results are presented in the table that follows. Trial Smith Walsh Trevor 1 11.1 19.0 14.6 2 13.5 18.0 15.7 3 15.3 19.8 16.8 4 14.6 19.6 16.7 5 9.8 16.6 15.2 -Referring to SCENARIO 10-4, based on the Tukey-Kramer procedure with an overall level of significance of 0.01, the agronomist would decide that there is a significant difference between the crop yield of Walsh and Trevor seeds.

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 computed t statistic is .

A Marine drill instructor recorded the time in which each of 11 recruits completed an obstacle course both before and after basic training.To test whether any improvement occurred, the instructor would use a t-distribution with 10 degrees of freedom.

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?" Some24% of the men compared to 62% of the women responded "Yes." Suppose that 150 women and200 men were interviewed.Construct a 99% 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 t Critical one-tail 1.685953 P ( T < t) two-tail 0.725009 t Critical two-tail 2.024394 -Referring to Scenario 10-13, suppose $\alpha$ = 0.05.Which of the following represents the result of the relevant hypothesis test?

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.Assuming W designates women's responses and M designates men's, which of the following are the appropriate null and alternative hypotheses to test the group's claim? a) $H _ { 0 } : \pi _ { M } - \pi _ { W } \leq 0$ versus $H _ { 1 } : \pi _ { M } - \pi _ { W } > 0$ b) $H _ { 0 } : \pi _ { M } - \pi _ { W } \geq 0$ versus $H _ { 1 } : \pi _ { M } - \pi _ { W } < 0$ c) $H _ { 0 } : \pi _ { M } - \pi _ { W } = 0$ versus $H _ { 1 } : \pi _ { M } - \pi _ { W } \neq 0$ d) $H _ { 0 } : \pi _ { M } - \pi _ { W } \neq 0 \leq 0$ versus $H _ { 1 } : \pi _ { M } - \pi _ { W } = 0$

An airline wants to select a computer software package for its reservation system.Four software packages (1, 2, 3, and 4) are commercially available.The airline will choose the package that bumps the fewest mean number of passengers as possible during a month.An experiment is setup in which each package is used to make reservations for 5 randomly selected weeks.(A total of20 weeks was included in the experiment.) The number of passengers bumped each week is given below.How should the data be analyzed? Package 1: $12,14,9,11,16$ Package 2: $2,4,7,3,1$ Package 3: $10,9,6,10,12$ Package 4: $7,6,6,15,12$

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, construct a 99% confidence interval estimate of the difference in proportion between the Kohler and Russell flash drives that are defective.

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 t Critical one-tail 1.685953 P ( T < t) two-tail 0.725009 t Critical two-tail 2.024394 -Referring to Scenario 10-13, which of the following represents the relevant hypotheses tested? a) $H _ { 0 } : \mu _ { I } - \mu _ { I I } \geq 0$ versus $H _ { 1 } : \mu _ { I } - \mu _ { I I } < 0$ b) $H _ { 0 } : \mu - \mu _ { I I } \leq 0$ versus $H _ { 1 } : \mu _ { I } - \mu _ { I I } > 0$ c) $H _ { 0 } : \mu - \mu _ { I I } = 0$ versus $H _ { 1 } : \mu _ { I } - \mu _ { I I } \neq 0$ d) $H _ { 0 } : \mu _ { I } - \mu _ { I I } \neq 0$ versus $H _ { 1 } : \mu _ { I } - \mu _ { I I } = 0$

SCENARIO 10-4 An agronomist wants to compare the crop yield of 3 varieties of chickpea seeds.She plants 15 fields, 5 with each variety.She then measures the crop yield in bushels per acre.Treating this as a completely randomized design, the results are presented in the table that follows. Trial Smith Walsh Trevor 1 11.1 19.0 14.6 2 13.5 18.0 15.7 3 15.3 19.8 16.8 4 14.6 19.6 16.7 5 9.8 16.6 15.2 -Referring to SCENARIO 10-4, construct the ANOVA table from the sample data.

SCENARIO 10-3 As part of an evaluation program, a sporting goods retailer wanted to compare the downhill coasting speeds of 4 brands of bicycles.She took 3 of each brand and determined their maximum downhill speeds.The results are presented in miles per hour in the table below. Trial Barth Tornado Reiser Shaw 1 43 37 41 43 2 46 38 45 45 3 43 39 42 46 -Referring to SCENARIO 10-3, what is the p-value of the test statistic for Levene's test for homogeneity of variances?

The t test for the difference between the means of 2 independent populations assumes that the respective

SCENARIO 10-2 A realtor wants to compare the mean sales-to-appraisal ratios of residential properties sold in four neighborhoods (A, B, C, and D).Four properties are randomly selected from each neighborhood and the ratios recorded for each, as shown below. A: 1.2,1.1,0.9,0.4 C: 1.0,1.5,1.1,1.3 B: 2.5,2.1,1.9,1.6 D: 0.8,1.3,1.1,0.7 Interpret the results of the analysis summarized in the following table: Source df SS MS F PR > F Neighborhoods 3.1819 1.0606 10.76 0.001 Error 12 Total 4.3644 -Referring to SCENARIO 10-2, the null hypothesis for Levene's test for homogeneity of variances is a) $H _ { 0 } : \mu _ { A } = \mu _ { B } = \mu _ { C } = \mu _ { D }$ b) $H _ { 0 } : M _ { A } = M _ { B } = M _ { C } = M _ { D }$ c) $H _ { 0 } : \sigma _ { A } ^ { 2 } = \sigma _ { B } ^ { 2 } = \sigma _ { C } ^ { 2 } = \sigma _ { D } ^ { 2 }$ d) $H _ { 0 } : \pi _ { A } = \pi _ { B } = \pi _ { C } = \pi _ { D }$

SCENARIO 10-2 A realtor wants to compare the mean sales-to-appraisal ratios of residential properties sold in four neighborhoods (A, B, C, and D).Four properties are randomly selected from each neighborhood and the ratios recorded for each, as shown below. A: 1.2,1.1,0.9,0.4 C: 1.0,1.5,1.1,1.3 B: 2.5,2.1,1.9,1.6 D: 0.8,1.3,1.1,0.7 Interpret the results of the analysis summarized in the following table: Source df SS MS F PR > F Neighborhoods 3.1819 1.0606 10.76 0.001 Error 12 Total 4.3644 -Referring to SCENARIO 10-2, the p-value of the test statistic for Levene's test for homogeneity of variances is

In testing for the differences between the means of 2 independent populations where the variances in each population are unknown but assumed equal, the degrees of freedom are

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 p-value of the test is _.

In testing for the differences between the means of two independent populations, you assume that the 2 populations each follow a distribution.

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 =106.4 hours =96.54 hours =10.3 hours =13.4 hours -Referring to Scenario 10-14, what is the critical value for testing if the population variances differ for preservatives I and II at the 5% level of significance?

SCENARIO 10-4 An agronomist wants to compare the crop yield of 3 varieties of chickpea seeds.She plants 15 fields, 5 with each variety.She then measures the crop yield in bushels per acre.Treating this as a completely randomized design, the results are presented in the table that follows. Trial Smith Walsh Trevor 1 11.1 19.0 14.6 2 13.5 18.0 15.7 3 15.3 19.8 16.8 4 14.6 19.6 16.7 5 9.8 16.6 15.2 -Referring to SCENARIO 10-4, the test is valid only if the population of crop yields has the same variance for the 3 varieties.

SCENARIO 10-3 As part of an evaluation program, a sporting goods retailer wanted to compare the downhill coasting speeds of 4 brands of bicycles.She took 3 of each brand and determined their maximum downhill speeds.The results are presented in miles per hour in the table below. Trial Barth Tornado Reiser Shaw 1 43 37 41 43 2 46 38 45 45 3 43 39 42 46 -Referring to SCENARIO 10-3, using an overall level of significance of 0.05, the critical value of the Studentized range Q used in calculating the critical range for the Tukey-Kramer procedure is_.