Yale University School of Management EMBA-MGT 511 Dec 1, 2007 Hypothesis Testing and Regression
Due: Dec 14, 2007. Email to: [email protected] with copy to [email protected]. Use the subject line MGT511-HW1.
Problem Set 1
1. The battery pack of a hand calculator is supposed to perform 20,000 calculations before needing recharging. The quality control manager for the manufacturer is concerned that the pack may not be working for as long as the specifications state. A test of 114 battery packs gives an average of 19,695 calculations and a standard deviation of 1103. a. Formulate the null and research hypothesis. b. Test the null hypothesis at alpha=0.05. (Write out all steps of the hypothesis testing procedure) c. Calculate the p-value.
2. A dealer in recycled paper places empty trailers at various sites; these are gradually filled by individuals who bring in old newspapers and the like. The trailers are picked up (and replaced by empties) on several schedules. One such schedule involves a pickup every second week. This schedule is desirable if the average amount of recycled paper is more than 1600 cubic feet per two week period. The dealer’s records for 18 two week periods show the following volumes (in cubic feet) at a particular site:
1660 1820 1590 1440 1730 1680 1750 1720 1900 1570 1700 1900 1800 1770 2010 1580 1620 1690 Assume that these figures represent the results of a random sample. a. Formulate the null and research hypothesis. (Specifically consider whether you should do a one-sided or two sided test). b. Test the null hypothesis at alpha=0.1 (Write out all steps of the hypothesis testing procedure) c. Calculate the p-value. 3. In 2000, 30 percent of all U.S. teenagers, aged 12 – 18, smoked. An advertising campaign was designed to reduce teenage smoking. To determine the effectiveness of the campaign, a random sample of U.S. teenagers is surveyed, one year after the advertising campaign started. Based on 1600 responses, the sample proportion smoking of teenagers (aged 11 – 17 in 2000) is found to be 28 percent.
a. Do you reject the null hypothesis that the population proportion equals 30 percent, at the five percent level? Note: If you use a one-tailed test, you must justify it. (Write out all steps of the hypothesis testing procedure). b. Describe in words (managerially interpretable) what the null and alternate hypotheses describe in this application. c. Is this a “fair test” of the effectiveness of the advertising campaign? Is 30 percent an appropriate value for the null hypothesis? Why or why not?
4. A team of builders has surveyed buyers of their new homes for years. Consistently, only 41% of the buyers have indicated they were “quite satisfied” or “very satisfied” with construction quality of their homes. The builders have adopted a revised quality inspection system to try to improve customer satisfaction. They have surveyed 104 buyers since then; these buyers seem representative, with no systematic changes from past purchasers. Of the 104 buyers, 51 indicated they were quite or very satisfied. a. Formulate the null hypothesis that there has been no real change in customer satisfaction from the past rate. b. Before taking such a survey, would you use a one-sided or two-sided research hypothesis? Explain. c. Calculate a z statistic for testing the null hypothesis. d. Show that the null hypothesis cannot be rejected, using a = 0.05.
5. A small manufacturer has a choice between shipping via the postal service and shipping via a private shipper. As a test, 10 destinations are chosen and packages are shipped to each by both routes. The delivery times in days are as follows: Postal Private Destination Service Shipper 1 3 2 2 4 2 3 5 3 4 4 5 5 8 4 6 9 6 7 7 9 8 10 6 9 9 7 10 9 6 a. Is a paired sample or independent sample test appropriate? Justify. b. Test the null hypothesis of no mean difference in delivery times against the research hypothesis that the private shipper has a shorter average delivery time. Use alpha = 0.05.
6. To understand your competitive position, you have examined the reliability of your product as well as those of your closest competitor’s product. You have subjected each product to abuse that represents a year’s worth of wear and tear per day. The table below shows the data indicating how long each item lasted.
Your Products Competitor’s products 1.0 0.2 8.9 2.8 1.2 1.7 10.3 7.2 4.9 2.2 1.8 2.5 3.1 2.6 3.6 2.0 2.1 0.5 2.9 2.3 8.6 1.9 5.3 1.2 6.6 0.5 1.2 a. Find the average time to failure for your and your competitor products. Find the average difference (between your and your competitors). b. Find the appropriate standard error for this average difference. In particular, is this a paired or unpaired situation? Why? c. Find the two-sided 95% confidence interval for the mean difference in reliability. d. Perform a hypothesis test at the 5% level of significance whether there is a difference between your products and your competitor’s. e. Find the p-value for the difference in the reliability. f. Write a brief paragraph, complete with footnote, that might be used in an advertising brochure showing off your products.