Compare the p-value from part a to the p-value obtained in Exercise 5.44. x

5.9 Inferences about the Median 5.52 Suppose we have a random sample of n ⫽ 15 measurements from a population having population median M. The research design calls for a 95 confidence interval on M.

a. Use Table 4 in the Appendix to obtain L

a 兾2 and U a 兾2 .

b. Use the large-sample approximation to determine L

a 兾2 and U a 兾2 . Compare these val- ues to the values obtained in part a. 5.53 Suppose we have a random sample of n ⫽ 45 measurements from a population having population median M. The research design calls for a 95 confidence interval on M.

a. Use Table 4 in the Appendix to obtain L

a 兾2 and U a 兾2 .

b. Use the large-sample approximation to determine L

a 兾2 and U a 兾2 . Compare these values to the values obtained in part a. 5.54 A researcher selects a random sample of 30 units from a population having a median M. Construct the rejection region for testing the research hypothesis H a : M ⬎ M using a ⫽ .01 and values in Table 4 of the Appendix. 5.55 Refer to Exercise 5.54. Use the large-sample approximation to set up the rejection region for testing the research hypothesis H a : M ⬎ M using a ⫽ .01. Compare this rejection region to the rejection region obtained in Exercise 5.54. Bus. 5.56 The amount of money spent on health care is an important issue for workers because many companies provide health insurance that only partial covers many medical procedures. The direc- tor of employee benefits at a midsize company wants to determine the amount spent on health care by the typical hourly worker in the company. A random sample of 25 workers is selected and the amount they spent on their families’ health care needs during the past year is given here. 400 345 248 1,290 398 218 197 342 208 223 531 172 4,321 143 254 201 3,142 219 276 326 207 225 123 211 108

a. Graph the data using a boxplot or normal probability plot and determine whether the

population has a normal distribution.

b. Based on your answer to part a, is the mean or the median cost per household a

more appropriate measure of what the typical worker spends on health care needs?

c. Place a 95 confidence interval on the amount spent on health care by the typical

worker. Explain what the confidence interval is telling us about the amount spent on health care needs. d. Does the typical worker spend more than 400 per year on health care needs? Use a ⫽ .05. Gov. 5.57 Many states have attempted to reduce the blood-alcohol level at which a driver is declared to be legally drunk. There has been resistance to this change in the law by certain business groups who have argued that the current limit is adequate. A study was conducted to demonstrate the effect on reaction time of a blood-alcohol level of .1, the current limit in many states. A ran- dom sample of 25 persons of legal driving age had their reaction time recorded in a standard labo- ratory test procedure before and after drinking a sufficient amount of alcohol to raise their blood alcohol to a .1 level. The difference After − Before in their reaction times in seconds was recorded as follows: .01 .02 .04 .05 .07 .09 .11 .26 .27 .27 .28 .28 .29 .29 .30 .31 .31 .32 .33 .35 .36 .38 .39 .39 .40 a. Graph the data and assess whether the population has a normal distribution. b. Place a 99 confidence interval on both the mean and median difference in reaction times of drivers who have a blood-alcohol level of .1.

c. Is there sufficient evidence that a blood-alcohol level of .1 causes any increase in

the mean reaction time?

d. Is there sufficient evidence that a blood-alcohol level of .1 causes any increase in

the median reaction time?

e. Which summary of reaction time differences seems more appropriate, the mean or

median? Justify your answer. 5.58 Refer to Exercise 5.57. The lobbyist for the business group has his expert examine the experimental equipment and determines that there may be measurement errors in recording the reaction times. Unless the difference in reaction time is at least .25 seconds, the expert claims that the two times are essentially equivalent.

a. Is there sufficient evidence that the median difference in reaction time is greater than

.25 seconds?

b. What other factors about the drivers are important in attempting to decide whether

moderate consumption of alcohol affects reaction time? Soc. 5.59 In an attempt to increase the amount of money people would receive at retirement from Social Security, the U.S. Congress during its 1999 session debated whether a portion of Social Security funds should be invested in the stock market. Advocates of mutual stock funds reassure the public by stating that most mutual funds would provide a larger retirement income than the income currently provided by Social Security. The annual rates of return of two highly recom- mended mutual funds for the years 1989 through 1998 are given here. The annual rate of return is defined as P 1 ⫺ P 兾P , where P and P 1 are the prices of the fund at the beginning and end of the year, respectively. Year 1989 1990 1991 1992 1993 1994 1995 1996 1997 1998 Fund A 25.4 17.1 ⫺ 8.9 26.7 3.6 ⫺ 8.5 ⫺ 1.3 32.9 22.9 26.6 Fund B 31.9 ⫺ 8.4 41.8 6.2 17.4 ⫺ 2.1 30.5 15.8 26.8 5.7

a. For both fund A and fund B, estimate the mean and median annual rate of return and

construct a 95 confidence interval for each.

b. Which of the parameters, the mean or median, do you think best represents the

annual rate of return for fund A and for fund B during the years 1989 through 1998? Justify your answer. 5.60 Refer to Exercise 5.59. a. Is there sufficient evidence that the median annual rate of return for the two mutual funds is greater than 10?

b. Is there sufficient evidence that the mean annual rate of return for the two mutual

funds is greater than 10? 5.61 What other summaries of the mutual fund’s rate of return are of importance to a person selecting a retirement plan? 5.62 Using the information in Table 5.8, answer the following questions. a. If the population has a normal distribution, then the population mean and median are identical. Thus, either the mean or median could be used to represent the center of the population. In this situation, why is the t test more appropriate than the sign test for testing hypotheses about the center of the distribution?

b. Suppose the population has a distribution that is highly skewed to the right. The

researcher uses an a ⫽ .05 t test to test hypotheses about the population mean. If the sample size n ⫽ 10, will the probability of a Type I error for the test be .05? Justify your answer.

c. When testing hypotheses about the mean or median of a highly skewed population, the

difference in power between the sign and t test decreases as the size of M a ⫺ M increases. Verify this statement using the values in Table 5.8. Why do think this occurs?

d. When testing hypotheses about the mean or median of a lightly skewed population,

the difference in power between the sign and t test is much less than that for a highly skewed population distribution. Verify this statement using the values in Table 5.8. Why do you think this occurs? Supplementary Exercises H.R. 5.63 An office manager has implemented an incentive plan that she thinks will reduce the mean time required to handle a customer complaint. The mean time for handling a complaint was 30 minutes prior to implementing the incentive plan. After the plan was in place for several