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

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 months, a random sample of the records of 38 customers who had complaints revealed a mean time of 28.7 minutes with a standard deviation of 3.8 minutes. a. Give a point estimate of the mean time required to handle a customer complaint. b. What is the standard deviation of the point estimate given in a?

c. Construct a 95 confidence on the mean time to handle a complaint after imple-

menting the plan. Interpret the confidence interval for the office manager.

d. Is there sufficient evidence that the incentive plan has reduced the mean time to handle

a complaint? Env. 5.64 The concentration of mercury in a lake has been monitored for a number of years. Mea- surements taken on a weekly basis yielded an average of 1.20 mgm 3 milligrams per cubic meter with a standard deviation of .32 mgm 3 . Following an accident at a smelter on the shore of the lake, 15 measurements produced the following mercury concentrations. 1.60 1.77 1.61 1.08 1.07 1.79 1.34 1.07 1.45 1.59 1.43 2.07 1.16 0.85 2.11 a. Give a point estimate of the mean mercury concentration after the accident. b. Construct a 95 confidence interval on the mean mercury concentration after the accident. Interpret this interval.

c. Is there sufficient evidence that the mean mercury concentration has increased since

the accident? Use a ⫽ .05.

d. Assuming that the standard deviation of the mercury concentration is .32 mgm

3 , calculate the power of the test to detect mercury concentrations of 1.28, 1.32, 1.36, and 1.40. Med. 5.65 Over the years, projected due dates for expectant mothers have been notoriously bad at a large metropolitan hospital. The physicians attended an in-service program to develop tech- niques to improve their projections. In a recent survey of 100 randomly selected mothers who had delivered a baby at the hospital since the in-service, the average number of days to birth beyond the projected due date was 9.2 days with a standard deviation of 12.4 days. a. Describe how to select the random sample of 100 mothers. b. Estimate the mean number of days to birth beyond the due date using a 95 confidence interval. Interpret this interval.

c. If the mean number of days to birth beyond the due date was 13 days prior to the in-

service, is there substantial evidence that the mean has been reduced? What is the level of significance of the test?

d. What factors may be important in explaining why the doctors’ projected due dates

are not closer to the actual delivery dates? Med. 5.66 In a standard dissolution test for tablets of a particular drug product, the manufacturer must obtain the dissolution rate for a batch of tablets prior to release of the batch. Suppose that the dissolution test consists of assays for 24 randomly selected individual 25 mg tablets. For each test, the tablet is suspended in an acid bath and then assayed after 30 minutes. The results of the 24 assays are given here. 19.5 19.7 19.7 20.4 19.2 19.5 19.6 20.8 19.9 19.2 20.1 19.8 20.4 19.8 19.6 19.5 19.3 19.7 19.5 20.6 20.4 19.9 20.0 19.8

a. Using a graphical display, determine whether the data appear to be a random sample

from a normal distribution.

b. Estimate the mean dissolution rate for the batch of tablets, for both a point estimate

and a 99 confidence interval.

c. Is there significant evidence that the batch of pills has a mean dissolution rate less

than 20 mg 80 of the labeled amount in the tablets? Use a ⫽ .01.

d. Calculate the probability of a Type II error if the true dissolution rate is 19.6 mg. Bus.

5.67 Statistics has become a valuable tool for auditors, especially where large inventories are in- volved. It would be costly and time consuming for an auditor to inventory each item in a large op- eration. Thus, the auditor frequently resorts to obtaining a random sample of items and using the sample results to check the validity of a company’s financial statement. For example, a hospital