If the level of confidence increases to 99.5 but the specified width of the interval

controlled apartments ranged from 200 to 1,500 per month. How many renters are needed in the survey to meet the requirements? 5.17 Refer to Exercise 5.16. Suppose the mayor has reviewed the proposed survey and decides on the following changes:

a. If the level of confidence is increased to 99 with the average rent estimated to

within 25, what sample size is required?

b. Suppose the budget for the project will not support both increasing the level of confi-

dence and reducing the width of the interval. Explain to the mayor the impact on the estimation of the average rent of not raising the level of confidence from 95 to 99. 5.4 A Statistical Test for ␮ 5.18 A researcher designs a study to test the hypotheses H : m ⱖ 28 versus H a : m ⬍ 28. A ran- dom sample of 50 measurements from the population of interest yields ⫽ 25.9 and s ⫽ 5.6.

a. Using a ⫽ .05, what conclusions can you make about the hypotheses based on the

sample information? b. Calculate the probability of making a Type II error if the actual value of m is at most 27. c. Could you have possibly made a Type II error in your decision in part a? Explain your answer.

5.19 Refer to Exercise 5.18. Sketch the power curve for rejecting H : m ⱖ 28 by determining

PWRm a for the following values of m: 22, 23, 24, 25, 26, and 27. a. Interpret the power values displayed in your graph. b. Suppose we keep n ⫽ 50 but change to a ⫽ .01. Without actually recalculating the values for PWRm a , sketch on the same graph as your original power curve, the new power curve for n ⫽ 50 and a ⫽ .01.

c. Suppose we keep a ⫽ .05 but change to n ⫽ 20. Without actually recalculating the

values for PWRm a , sketch on the same graph as your original power curve the new power curve for n ⫽ 20 and a ⫽ .05. 5.20 Use a computer software program to simulate 100 samples of size 25 from a normal distri- bution with m = 30 and s ⫽ 5. Test the hypotheses H : m ⫽ 30 versus H a : using each of the 100 samples of n ⫽ 25 and using a ⫽ .05.

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

reject H ?

b. Suppose you were to conduct 100 tests of hypotheses and in each of these tests the

true hypothesis was H . On the average, how many of the 100 tests would have re- sulted in your incorrectly rejecting H , if you were using a ⫽ .05? c. What type of error are you making if you incorrectly reject H ? 5.21 Refer to Exercise 5.20. Suppose the population mean was 32 instead of 30. Simulate 100 samples of size n ⫽ 25 from a normal distribution with m ⫽ 32 and s ⫽ 5. Using a ⫽ .05, test the hypotheses H : m ⫽ 30 versus H a : using each of the 100 samples of size n ⫽ 25.

a. What proportion of the 100 tests of hypotheses resulted in the correct decision, that

is, reject H ?

b. In part a, you were estimating the power of the test when m

a ⫽ 32, that is, the ability of the testing procedure to detect that the null hypothesis was false. Now, calculate the power of your test to detect that m ⫽ 32, that is, compute PWRm a ⫽ 32.

c. Based on your calculation in b how many of the 100 tests of hypotheses would you

expect to correctly reject H ? Compare this value with the results from your simulated data. 5.22 Refer to Exercises 5.20 and 5.21. a. Answer the questions posed in these exercises with a ⫽ .01 in place of a ⫽ .05. You can use the data set simulated in Exercise 5.20, but the exact power of the test, PWRm a ⫽ 32, must be recalculated. b. Did decreasing a from .05 to .01 increase or decrease the power of the test? Explain why this change occurred. Med. 5.23 A study was conducted of 90 adult male patients following a new treatment for congestive heart failure. One of the variables measured on the patients was the increase in exercise capacity m ⫽ 30 m ⫽ 30 y 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