Exam 10: Two-Sample Tests

When you test for differences between the means of two independent populations, you can only use a two-tail test.

SCENARIO 10-5 A hotel chain has identically small sized resorts in 5 locations in different small islands.The data that follow resulted from analyzing the hotel occupancies on randomly selected days in the 5 locations. ROW Location A Location B Location C Location D Location E 1 28 40 21 37 22 2 33 35 21 47 19 3 41 33 27 45 25 Analysis of Variance Source df SS MS F p Location 4 963.6 11.47 0.001 Error 10 210.0 Total -Referring to SCENARIO 10-5, the total mean squares is 261.90.

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, the test is valid only if the population of speeds has the same variance for the 4 brands.

When would you use the Tukey-Kramer procedure?

If you 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 _.

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 10-3 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, based on the Tukey-Kramer procedure with an overall level of significance of 0.05, the retailer would decide that there is a significant difference between all pairs of mean speeds.

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.

When the sample sizes are equal, the pooled variance of the two groups is the average of the 2 sample variances.

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 among-group variation or SSA 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 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, construct a 99% confidence interval estimate of the difference in proportion of workers who would like to attend a self-improvement course in the recent study and the past study.

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.

In testing for the differences between the means of two related populations, thehypothesis is the hypothesis of "no differences."

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 BeforeCourse(1) Exam Score AfterCourse(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, in examining the differences between related samples we are essentially sampling from an underlying population of difference "scores."

The F test statistic in a one-way ANOVA is

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, what is the largest level of significance at which a test on a difference in the variabilities of the amount of time required to reach a customer service representative between the two hotels will not be rejected?

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 _.

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.What is the value of the test statistic?

SCENARIO 10-2 A researcher randomly sampled 30 graduates of an MBA program and recorded data concerning their starting salaries.Of primary interest to the researcher was the effect of gender on starting salaries. The result of the pooled-variance t-test of the mean salaries of the females (Population 1) and males (Population 2) in the sample is given below. Hypothesized Difference 0 Level of Significance 0.05 Population 1 Sample Sample Size 18 Sample Mean 99210 Sample Standard Deviation 13577 Population 2 Sample Sample Size 12 Sample Mean 105820 Sample Standard Deviation 11741 Difference in Sample Means -6610 t Test Statistic -1.37631 Lower-Tail Test Lower Critical Value -1.70113 p-Value 0.089816 -Referring to Scenario 10-2, the researcher was attempting to show statistically that the female MBA graduates have a significantly lower mean starting salary than the male MBA graduates.The proper conclusion for this test is:

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 is normally distributed for the 3 varieties.