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

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

a. Estimate the average amount of SO

2 emissions during each of the three time periods using 95 confidence intervals.

b. Does there appear to be a significant difference in average SO

2 emissions over the three time periods?

c. Combining the data over the entire day, is the average SO

2 emissions using the new scrubber less than .145, the average daily value for the old scrubber? Soc. 5.72 As part of an overall evaluation of training methods, an experiment was conducted to de- termine the average exercise capacity of healthy male army inductees. To do this, each male in a random sample of 35 healthy army inductees exercised on a bicycle ergometer a device for measuring work done by the muscles under a fixed work load until he tired. Blood pressure, pulse rates, and other indicators were carefully monitored to ensure that no one’s health was in danger. The exercise capacities mean time, in minutes for the 35 inductees are listed here. 23 19 36 12 41 43

19 28

14 44 15 46 36 25 35 25 29 17 51 33 47 42 45 23 29 18 14 48 21 49

27 39

44 18 13

a. Use these data to construct a 95 confidence interval for m, the average exercise

capacity for healthy male inductees. Interpret your findings. b. How would your interval change using a 99 confidence interval? 5.73 Using the data in Exercise 5.72, determine the number of sample observations that would be required to estimate m to within 1 minute, using a 95 confidence interval. Hint: Substitute s ⫽ 12.36 for s in your calculations. H.R. 5.74 Faculty members in a state university system who resign within 10 years of initial em- ployment are entitled to receive the money paid into a retirement system, plus 4 per year. Un- fortunately, experience has shown that the state is extremely slow in returning this money. Concerned about such a practice, a local teachers’ organization decides to investigate. From a random sample of 50 employees who resigned from the state university system over the past 5 years, the average time between the termination date and reimbursement was 75 days, with a standard deviation of 15 days. Use the data to estimate the mean time to reimbursement, using a 95 confidence interval. 5.75 Refer to Exercise 5.74. After a confrontation with the teachers’ union, the state promised to make reimbursements within 60 days. Monitoring of the next 40 resignations yields an aver- age of 58 days, with a standard deviation of 10 days. If we assume that these 40 resignations rep- resent a random sample of the state’s future performance, estimate the mean reimbursement time, using a 99 confidence interval. Bus. 5.76 Improperly filled orders are a costly problem for mail-order houses. To estimate the mean loss per incorrectly filled order, a large firm plans to sample n incorrectly filled orders and to determine the added cost associated with each one. The firm estimates that the added cost is between 40 and 400. How many incorrectly filled orders must be sampled to estimate the mean additional cost using a 95 confidence interval of width 20? Eng. 5.77 The recipe for producing a high-quality cement specifies that the required percentage of SiO 2 is 6.2. A quality control engineer evaluates this specification weekly by randomly selecting samples from n ⫽ 20 batches on a daily basis. On a given day, she obtained the following values: 1.70 9.86 5.44 4.28 4.59 8.76 9.16 6.28 3.83 3.17 5.98 2.77 3.59 3.17 8.46 7.76 5.55 5.95 9.56 3.58

a. Estimate the mean percentage of SiO

2 using a 95 confidence interval.

b. Evaluate whether the percentage of SiO

2 is different from the value specified in the recipe using an a ⫽ .05 test of hypothesis.

c. Use the following plot to determine if the procedures you used in parts a and b

were valid.