How many of the 100 tests of hypotheses resulted in your reaching the decision to

in minutes over a 4-week treatment period. The previous treatment regime had produced an average increase of m ⫽ 2 minutes. The researchers wanted to evaluate whether the new treat- ment had increased the value of m in comparison to the previous treatment. The data yielded ⫽ 2.17 and s ⫽ 1.05. a. Using a ⫽ .05, what conclusions can you draw about the research hypothesis? b. What is the probability of making a Type II error if the actual value of m is 2.1?

5.24 Refer to Exercise 5.23. Compute the power of the test PWRm

a at m a ⫽ 2.1, 2.2, 2.3, 2.4, and 2.5. Sketch a smooth curve through a plot of PWRm a versus m a . a. If a is reduced from .05 to .01, what would be the effect on the power curve? b. If the sample size is reduced from 90 to 50, what would be the effect on the power curve? 5.5 Choosing the Sample Size for Testing ␮ Med. 5.25 A national agency sets recommended daily dietary allowances for many supplements. In particular, the allowance for zinc for males over the age of 50 years is 15 mgday. The agency would like to determine if the dietary intake of zinc for active males is significantly higher than 15 mgday. How many males would need to be included in the study if the agency wants to construct an

a ⫽

.05 test with the probability of committing a Type II error to be at most .10 whenever the average zinc content is 15.3 mgday or higher? Suppose from previous studies they estimate the standard deviation to be approximately 4 mgday. Edu. 5.26 To evaluate the success of a 1-year experimental program designed to increase the mathe- matical achievement of underprivileged high school seniors, a random sample of participants in the program will be selected and their mathematics scores will be compared with the previous year’s statewide average of 525 for underprivileged seniors. The researchers want to determine whether the experimental program has increased the mean achievement level over the previous year’s statewide average. If a ⫽ .05, what sample size is needed to have a probability of Type II error of at most .025 if the actual mean is increased to 550? From previous results, .

5.27 Refer to Exercise 5.26. Suppose a random sample of 100 students is selected yielding

⫽ 542 and s ⫽ 76. Is there sufficient evidence to conclude that the mean mathematics achieve- ment level has been increased? Explain. Bus. 5.28 The administrator of a nursing home would like to do a time-and-motion study of staff time spent per day performing nonemergency tasks. Prior to the introduction of some efficiency measures, the average person-hours per day spent on these tasks was m ⫽ 16. The administrator wants to test whether the efficiency measures have reduced the value of m. How many days must be sampled to test the proposed hypothesis if she wants a test having a ⫽ .05 and the probability of a Type II error of at most .10 when the actual value of m is 12 hours or less at least a 25 decrease from prior to the efficiency measures being implemented? Assume s ⫽ 7.64. Env. 5.29 The vulnerability of inshore environments to contamination due to urban and industrial expansion in Mombasa is discussed in the paper “Metals, petroleum hydrocarbons and organo- chlorines in inshore sediments and waters on Mombasa, Kenya” Marine Pollution Bulletin, 1997, pp. 570 –577. A geochemical and oceanographic survey of the inshore waters of Mombasa, Kenya, was undertaken during the period from September 1995 to January 1996. In the survey, suspended particulate matter and sediment were collected from 48 stations within Mombasa’s estuarine creeks. The concentrations of major oxides and 13 trace elements were determined for a varying number of cores at each of the stations. In particular, the lead concentrations in suspended particulate matter mg kg ⫺ 1 dry weight were determined at 37 stations. The researchers were interested in determining whether the average lead concentration was greater than 30 mg kg ⫺ 1 dry weight. The data are given in the following table along with summary statistics and a normal probability plot. Lead concentrations mg kg ⫺ 1 dry weight from 37 stations in Kenya 48 53 44 55 52 39 62 38 23 27 41 37 41 46 32 17 32 41 23 12 3 13 10 11 5 30 11 9 7 11 77 210 38 112 52 10 6 y s ⬇ 80 y