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www.downloadslide.com Exercises 539 Runners Cyclists Condition Mean s Mean s Rest 14.5 3.92 11.1 3.98 80% maximal O 2 12.2 3.49 11.5 4.95 consumption Maximal O 2 consumption 19.1 16.9 12.2 4.67 a Is there sufficient evidence to support a claim that the variability of compartment pressure differs for runners and cyclists who are resting? Use α = .05. b i What can be said about the attained significance level using a table in the appendix? ii Applet Exercise What can be said about the attained significance level using the appropriate applet? c Is there sufficient evidence to support a claim that the variability in compartment pressure between runners and cyclists differs at maximal O 2 consumption? Use α = .05. d i What can be said about the attained significance level using a table in the appendix? ii Applet Exercise What can be said about the attained significance level using the appropriate applet? 10.83 The manager of a dairy is in the market for a new bottle-filling machine and is considering machines manufactured by companies A and B. If ruggedness, cost, and convenience are comparable in the two machines, the deciding factor will be the variability of fills (the machine producing fills **with** the smaller variance being preferable). Let σ1 2 and σ 2 2 be the fill variances for machines produced by companies A and B, respectively. Now consider various tests of the null hypothesis H 0 : σ1 2 = σ 2 2 . Obtaining samples of fills from the two machines and using the test statistic S1 2/S2 2 , we could set up as the rejection region an upper-tail area, a lower-tail area, or a two-tailed area of the F distribution, depending on the interests to be served. Identify the type of rejection region that would be most favored by the following persons, and explain why. a b c The manager of the dairy A salesperson for company A A salesperson for company B 10.84 An experiment published in The American Biology Teacher studied the efficacy of using 95% ethanol and 20% bleach as disinfectants for removing bacterial and fungal contamination when culturing plant tissues. The experiment was repeated 15 times **with** each disinfectant, using eggplant as the plant tissue cultured. 17 Five cuttings per plant were placed on a petri dish, disinfected using each agent, and stored at 25 ◦ C for 4 weeks. The observations reported were the number of uncontaminated eggplant cuttings after the 4 weeks of storage. Relevant data is given in the following table. Are you willing to assume that the underlying population variances are equal? Disinfectant 95% Ethanol 20% Bleach Mean 3.73 4.80 Variance 2.78095 0.17143 n 15 15 17. Source: Michael Brehm, J. Buguliskis, D. Hawkins, E. Lee, D. Sabapathi, and R. Smith, “Determining Differences in Efficacy of Two Disinfectants Using t tests,” The American Biology Teacher 58(2), (1996): 111.

www.downloadslide.com 540 Chapter 10 Hypothesis Testing a What can be said about the attained significance level using the F table in the appendix? b Applet Exercise What can be said about the attained significance level using the applet F-Ratio Probabilities and Quantiles? c What would you conclude, **with** α = .02? 10.85 Applet Exercise A precision instrument is guaranteed to be accurate to **with**in 2 units. A sample of four instrument readings on the same object yielded the measurements 353, 351, 351, and 355. Give the attained significance level for testing the null hypothesis σ = .7 versus the alternative hypothesis σ>.7. 10.86 Aptitude tests should produce scores **with** a large amount of variation so that an administrator can distinguish between persons **with** low aptitude and persons **with** high aptitude. The standard test used by a certain industry has been producing scores **with** a standard deviation of 10 points. A new test is given to 20 prospective employees and produces a sample standard deviation of 12 points. Are scores from the new test significantly more variable than scores from the standard? Use α = .01. 10.87 Refer to Exercise 10.70. Is there sufficient evidence, at the 5% significance level, to support concluding that the variance in measurements of DDT levels is greater for juveniles than it is for nestlings? 10.10 Power of Tests and the Neyman–Pearson Lemma In the remaining sections of this chapter, we move from practical examples of statistical tests to a theoretical discussion of their properties. We have suggested specific tests for a number of practical hypothesis testing situations, but you may wonder why we chose those particular tests. How did we decide on the test statistics that were presented, and how did we know that we had selected the best rejection regions? The goodness of a test is measured by α and β, the probabilities of type I and type II errors, respectively. Typically, the value of α is chosen in advance and determines the location of the rejection region. A related but more useful concept for evaluating the performance of a test is called the power of the test. Basically, the power of a test is the probability that the test will lead to rejection of the null hypothesis. DEFINITION 10.3 Suppose that W is the test statistic and RR is the rejection region for a test of a hypothesis involving the value of a parameter θ. Then the power of the test, denoted by power(θ), is the probability that the test will lead to rejection of H 0 when the actual parameter value is θ. That is, power(θ) = P(W in RR when the parameter value is θ). Suppose that we want to test the null hypothesis H 0 : θ = θ 0 and that θ a is a particular value for θ chosen from H a . The power of the test at θ = θ 0 , power(θ 0 ),is equal to the probability of rejecting H 0 when H 0 is true. That is, power(θ 0 ) = α, the probability of a type I error. For any value of θ from H a , the power of a test measures

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