SECTION 6.5 Exercises
Question 6.108
6.108 Make a recommendation.
Your manager has asked you to review a research proposal that includes a section on sample size justifcation. A careful reading of this section indicates that the power is 20% for detecting an effect that you would consider important. Write a short report for your manager explaining what this means, and make a recommendation on whether or not this study should be run.
Question 6.109
6.109 Explain power and sample size.
Two studies are identical in all respects except for the sample sizes. Consider the power versus a particular sample size. Will the study with the larger sample size have more power or less power than the one with the smaller sample size? Explain your answer in terms that could be understood by someone with very little knowledge of statistics.
6.109
The one with the larger sample size will have more power. More data will provide more information, which should give us a better chance of finding differences between the data and the hypothesis.
Question 6.110
6.110 Power versus a different alternative.
The power for a two-sided test of the null hypothesis versus the alternative is 0.73. What is the power versus the alternative ? Draw a picture and use this to explain your answer.
Question 6.111
6.111 Power versus a different alternative.
A one-sided test of the null hypothesis versus the alternative has power equal to 0.5. Will the power for the alternative be higher or lower than 0.5? Draw a picture and use this to explain your answer.
351
6.111
Higher because it is farther away from 60.
Question 6.112
6.112 Effect of changing the alternative on power.
The Statistical Power applet illustrates the power calculation similar to that in Figure 6.18 (page 345). Open the applet and keep the default settings for the null () and the alternative () hypotheses, the sample size (), the standard deviation (), and the significance level (). In the “alt ” box, enter the value 1. What is the power? Repeat for alternative equal to 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9. Make a table giving and the power. What do you conclude?
Question 6.113
6.113 Decreasing population standard deviation.
Improved measurement systems, better technology, and changes to standard operating procedures are among various strategies to reduce population variability in manufacturing and service applications. Suppose variation reduction strategies are implemented and reduce the population standard deviation by 50%; that is, it is half of its original value.
- If is the sample size used for hypothesis testing under the original standard deviation, what sample size in terms of is now required to maintain some specified power?
- If the new sample size were used, what might you be concerned about? (Hint: Think about the shape of the sampling distribution.)
6.113
(a) 0.25n. (b) Smaller samples have more variation, so even when the population standard deviation was reduced, our sample standard deviation may be bigger and reduce our power.
Question 6.114
6.114 Sample size determination.
CASE 6.1 Example 6.26 (page 343) considers the test of against , where is the mean fill amount. The population standard deviation is given to be . Suppose that the testing is performed at a 5% significance level. Without use of software, determine the sample size that is minimally required to give at least 0.8 power.
Question 6.115
6.115 Power of the mean blood pressure.
Example 6.20 (pages 327–328) gives a test of a hypothesis about systolic blood pressure of company executives based on a sample size of 72. The hypotheses are
Assume that the population standard deviation is . Consider the test at the 1% level of significance, which implies that it would reject when , where
Is this test sufficiently sensitive to usually detect a company mean blood pressure level of 133 with at least 0.8 power?
6.115
No, the power is only 0.6915.
Question 6.116
6.116 Choose the appropriate distribution.
You must decide which of two discrete distributions a random variable has. We call the distributions and . Here are the probabilities that the distributions assign to the values of :
| 0 | 1 | 2 | 3 | 4 | 5 | 6 | |
|---|---|---|---|---|---|---|---|
| 0.1 | 0.1 | 0.1 | 0.1 | 0.2 | 0.1 | 0.3 | |
| 0.2 | 0.1 | 0.1 | 0.2 | 0.2 | 0.1 | 0.1 |
You have a single observation on and wish to test
One possible decision procedure is to accept if or and reject otherwise.
- Find the probability of a Type I error; that is, the probability that you reject when is the correct distribution.
- Find the probability of a Type II error.
Question 6.117
6.117 Computer-assisted career guidance systems.
A wide variety of computer-assisted career guidance systems have been developed over the past decade. These programs use factors such as student interests, aptitude, skills, personality, and family history to recommend a career path. For simplicity, suppose that a program recommends a high school graduate either go to college or join the workforce.
- What are the two hypotheses and the two types of error that the program can make?
- The program can be adjusted to decrease one error probability at the cost of an increase in the other error probability. Which error probability would you choose to make smaller, and why? (This is a matter of judgment. There is no single correct answer.)
6.117
(a) Hypotheses: “go to college” and “join workforce.” (b) Errors: recommending “go to college” for someone that should “join the workforce” and recommending “join the workforce” for someone that should “go to college.”