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

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 financial statement claims an inventory that averages 300 per item. An auditor’s random sample of 20 items yielded a mean and standard deviation of 160 and 90, respectively. Do the data con- tradict the hospital’s claimed mean value per inventoried item and indicate that the average is less than 300? Use a ⫽ .05. Bus. 5.68 Over the past 5 years, the mean time for a warehouse to fill a buyer’s order has been 25 min- utes. Officials of the company believe that the length of time has increased recently, either due to a change in the workforce or due to a change in customer purchasing policies. The processing time in minutes was recorded for a random sample of 15 orders processed over the past month. 28 25

27 31

10 26

30 15 55 12

24 32

28 42 38 Do the data present sufficient evidence to indicate that the mean time to fill an order has increased? Engin. 5.69 If a new process for mining copper is to be put into full-time operation, it must produce an average of more than 50 tons of ore per day. A 15-day trial period gave the results shown in the accompanying table. Day 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Yield tons 57.8 58.3 50.3 38.5 47.9 157.0 38.6 140.2 39.3 138.7 49.2 139.7 48.3 59.2 49.7

a. Estimate the typical amount of ore produced by the mine using both a point estimate

and a 95 confidence interval.

b. Is there significant evidence that on a typical day the mine produces more than

50 tons of ore? Test by using a ⫽ .05. Env. 5.70 The board of health of a particular state was called to investigate claims that raw pollu- tants were being released into the river flowing past a small residential community. By applying financial pressure, the state was able to get the violating company to make major concessions to- ward the installation of a new water purification system. In the interim, different production sys- tems were to be initiated to help reduce the pollution level of water entering the stream. To monitor the effect of the interim system, a random sample of 50 water specimens was taken throughout the month at a location downstream from the plant. If ⫽ 5.0 and s ⫽ .70, use the sample data to determine whether the mean dissolved oxygen count of the water in ppm is less than 5.2, the average reading at this location over the past year. a. List the five parts of the statistical test, using a ⫽ .05. b. Conduct the statistical test and state your conclusion. Env. 5.71 The search for alternatives to oil as a major source of fuel and energy will inevitably bring about many environmental challenges. These challenges will require solutions to problems in such areas as strip mining and many others. Let us focus on one. If coal is considered as a major source of fuel and energy, we will have to consider ways to keep large amounts of sulfur dioxide SO 2 and particulates from getting into the air. This is especially important at large government and industrial operations. Here are some possibilities.

1. Build the smokestack extremely high. 2. Remove the SO

2 and particulates from the coal prior to combustion.

3. Remove the SO

2 from the gases after the coal is burned but before the gases are released into the atmosphere. This is accomplished by using a scrubber. A new type of scrubber has been recently constructed and is set for testing at a power plant. Over a 15-day period, samples are obtained three times daily from gases emitted from the stack. The amounts of SO 2 emissions in pounds per million BTU are given here: Day Time 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 6 A . M . .158 .129 .176 .082 .099 .151 .084 .155 .163 .077 .116 .132 .087 .134 .179 2 P . M . .066 .135 .096 .174 .179 .149 .164 .122 .063 .111 .059 .118 .134 .066 .104 10 P . M . .128 .172 .106 .165 .163 .200 .228 .129 .101 .068 .100 .119 .125 .182 .138 y