MORE HYPOTHESIS TESTING EXERCISES 1. WestWays magazine sampled random flights from LAX and George Bush Airport (IAH - Houston) to compare their rate of delayed departures. For LAX, 205 of the 845 flights sampled were delayed; for IAH, 312 out of 1296. Test at = 0.05 whether these samples indicate a different delayed departure rate for these airports. Do the test by hand (using Minitab just for the p-value) and confirm your results with Minitab. Interpret your results and state your conclusion in terms of the data. 2. Merrill Sachs asked major customers to rank their 2 newest financial consultants. The results are summarized below: Consultant A: Avg. rating 6.72/7 by 26 customers. Sample std. deviation = 0.60 Consultant B: Avg. rating 6.28/7 by 18 customers. Sample std. deviation = 0.77 a. What assumption(s) must be made for a 2-sample t test of the difference in the ratings to be valid? b. Making that assumption, conduct a 2-sample t test at = 0.05 of the difference in ratings. Do the test by hand (using Minitab just for the p-value) and confirm your results with Minitab. (The degrees of freedom for this test = 16.) Interpret your results and state your conclusion in the terms of the data. c. Suppose that, instead of 26 and 18 the sample sizes were 260 and 180, respectively. Repeat the test using Minitab. Interpret your results and state your conclusion in terms of the data. a. b. c. d. 3. A marketing researcher decides to test whether coffee drinkers have a favorable view of coffee labeled as \"Fair Trade.\" He has 15 randomly selected coffee drinkers taste his company's Morning Blend, then the exact same coffee labeled Fair Trade Blend. They are asked to rate the coffees on a 1 to 10 scale, 10 being best. The data is recorded in Minitab and a statistical test is done at = 0.05 to determine if \"Fair Trade\" coffee is rated more highly. What is the correct hypothesis test for testing this data? What condition(s), if any, are necessary for the test to be valid? What are the hypotheses for this test? Open the FairTrade dataset in Minitab and use Minitab to conduct the test. Assume all conditions exist for a valid test. Interpret your results and state your conclusion in terms of the data. 1 4. We wish to test the variances of 2 populations to assess if we can reasonably assume they are equal. We take independent random samples of size 50 from each. Histograms of each sample are shown below. Would the F test be the appropriate test to use for these populations? Why or why not? 5. Suppose we perform a valid 1-sided F-test on the populations of male and female Angus beef cattle. The sample of males has a higher standard deviation. We correctly compute the p-value to be 0.028. If we are doing this F-test at = 0.05, what is our conclusion? 6. One version of the Mann-Whitney U test is a hypothesis test used to compare the medians of two populations. What is the null hypothesis of this version of the Mann-Whitney U test? 7. Open the PurdueIQ dataset in Minitab. IQ's are recorded for 36 randomly selected Purdue undergrads. Use Minitab to do an Anderson-Darling (AD) test at the 5% level to test whether the IQ's of Purdue undergrads is normally distributed. Then test at the 5% level whether the std. dev. of Purdue undergrad IQ's is different from the overall population std. dev. of 15. (Hint: Given the results of your AD test, which test of std. dev. should you use?) Copy and paste or attach all relevant Minitab output. Interpret your results and state your conclusion in terms of the data. 2 8. A bolt manufacturer is using a hypothesis test with = 0.01 to see if the diameter of their 0.75 cm diameter bolts are being manufactured properly. The goal is to have the average bolt diameter be 0.75 0.0075 cms. (Thus, a 2-sided test with difference of 0.0075 is used.) They know that the standard deviation of the 0.75 cm bolts is 0.028 cms. Use Minitab to compute the sample size necessary to achieve a power of 0.9. According to the Power Curve, if a difference of 0.0025 must be used, what is the approximate power of the test using this sample size? Copy and paste or attach your Minitab output to your paper. 9. In a contentious shareholder lawsuit, a critical claim by the plaintiffs is that average CEO tenure for companies of at least $2 billion in market cap is 9 years. A survey of 30 such companies had a sample mean of 7.87 with a sample standard deviation of 6.28. Use this sample to test at the 5% level whether the average CEO tenure is less than 9 years. Do the test by hand (using Minitab just for the p-value) and confirm your results with Minitab. Interpret your results and state your conclusion in terms of the data. 10. Assume that we randomly sample 300 residents of Indiana and ask them if they favor having the government issue marriage licenses to same-sex couples. 159 say that the do; 141 don't. Test at the 5% level whether the population support for this proposition is greater than 50%. Do the test by hand (using Minitab just for the p-value) and confirm your results with Minitab. Interpret your results and state your conclusion in terms of the data. 11. A flower shop wishes to add the valuable Waimea orchid to its product list. They purchase a large shipment of bulbs from a supplier in Kauai. It is established by Mendelian theory that the predominant colors in the Waimea orchid (blue, red, violet, orange) will occur with the ratio of 6:4:3:2. When the first 60 Waimea orchid bulbs bloom, the predominant colors are 27 blue, 10 red, 17 violet, 6 orange. The florists are concerned that these bulbs are not Waimea orchids, but a similar appearing (and more common, less valuable) hybrid. What hypothesis test can be used to test this? Conduct the test at = 0.05 using Minitab. Interpret your results and state your conclusion in terms of the data. 12. Drug clinical trials are analyzed using hypothesis testing methods. As we've discussed, the null hypothesis in the effective stage of such trials is that the drug being tested is ineffective. The alternative, of course, is that the drug is effective. Explain why such tests are usually done with low -levels. 3 13. An HR manager at a large agricultural equipment company is considering proposing putting the sales staff on 4 day - 10 hours/day work weeks, instead of their current 5 day - 8 hours/day week. It is believed that this will decrease the amount of driving they must do during a typical work week. They put 16 of their sales people on the 4 day week for one month, then on the 5 day week for one month, and have them record their driving distances. The data is recorded in the DrivingDistance dataset. Assume that all conditions for valid tests are met. a. Test at the 5% level whether the 4 day work week shortens the driving distance. State the result of this test in terms of the data. Consider the confidence bound associated with this test. Does it influence the decision to go to the 4-day week? Why or why not? b. Compute the power of this test with a difference of 500. Does it influence the decision? c. Do a 2-sided test at the 5% level to test whether there is a statistically significant difference in the driving distances. What is the probability of Type II error with this test? Does this test influence the decision to go to the 4-day week? What factors other than the data presented might the manager want to consider? a. b. c. d. e. 14. Statistical significance, Power, and Practical significance - A Big Data Issue: As a marketing associate for a sports apparel company, we see that, in 2011, 44.2% of our customers checked reviews of our new products on social media sites before buying. This year, to see if that percentage has increased, we use an email campaign to survey our recent customers. Of 39,994 respondents, 17,885 said that, before buying, they checked reviews of the product they purchased on social media. Based on this survey, what is the point estimate of the proportion of our customers who check social media reviews? Test whether the increase indicated by this point estimate is statistically significant? Compute a 95% confidence interval for the percentage of customers using social media before buying. What do we think about it? What is the power of the test in part b? Based on this analysis, would you recommend any changes in our social media marketing approach? 4 15. We wish to compare the variances of two populations. We take random samples from each with the following results: Sample 1: n = 16, sample standard deviation = 9.9 Sample 2: n = 22, sample standard deviation = 6.6 a. What are the hypotheses for a 1-sided F test? b. What is the F statistic for this test? c. The critical value for a 1-sided F test based on these samples at = 0.10 is as shown the graph below. Give the conclusion of this test at the 10% level. (Assume an Anderson-Darling test fails to reject the null hypothesis of normality and we can consider our F test results valid.) Distribution Plot F, df1=15, df2=21 0.9 0.8 0.7 Density 0.6 0.5 0.4 0.3 0.2 0.1 0.0 0.1 0 1.827 X 5 16. Blotto Brewing Co. has research from 3 years ago that shows that 17% of their customers prefer Blotto Dark, 45% prefer Blotto Lite, and 38% prefer Blotto Lager. To see if these percentages are still correct, they sample 125 regular Blotto drinkers. They enter their preferences in Minitab and perform an appropriate hypothesis test of the data. The results are displayed below. Chi-Square Goodness-of-Fit Test for Observed Counts in Variable: C3 Category Dark Lite Lager N 125 DF 2 Observed 19 71 35 Chi-Sq 7.39549 Test Proportion 0.17 0.45 0.38 Expected 21.25 56.25 47.50 Contribution to Chi-Sq 0.23824 3.86778 3.28947 P-Value 0.025 Which of the statements below is an accurate interpretation of the above test results? a. There appears to have been a significant shift in preferences from Lager to Lite. b. We have statistically significant evidence at the 5% level that the percentages from 3 years ago are no longer valid. c. We have statistically significant evidence at the 5% level that the percentages from 3 years ago are still valid. d. Both a and c. e. Both a and b. 6