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9-91. Suppose that we wish to test H0: 0 versus H1: 0 , where the population is normal with known . Let 0 , and de ne the critical region so that we will reject H0 if z0 z or if z0 z , where z0 is the value of the usual test statistic for these hypotheses. (a) Show that the probability of type I error for this test is . (b) Suppose that the true mean is 1 . Derive 0 an expression for for the above test. 9-92. Derive an expression for for the test on the variance of a normal distribution. Assume that the twosided alternative is speci ed. 9-93. When X1, X2, p , Xn are independent Poisson random variables, each with parameter , and n is large, the sample mean X has an approximate normal distribution with mean and variance n. Therefore, Z X 1 n 9-94. When X1, X2, p , Xn is a random sample from a normal distribution and n is large, the sample standard deviation has approximately a normal distribution with mean and variance 2 12n2 . Therefore, a large-sample test for H0: 0 can be based on the statistic 2 S

2. Choose fast, medium and slow switches from L , M , and R: Since these can tie selected in any order, there are 6 = 22.6 combinations.

2 0 0

12n2

Be aware that not all Menu tabs and items are available at all times. If you can t locate a Menu tab or item, try putting the camera into another Shooting mode by turning the Mode dial or pressing the Playback button.

3 . Select input and output plugboard permutations: If the input and outjput plugboards can be chosen independently, there arc (26!) = 21763.8 combinations.

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has approximately a standard normal distribution. Thus we can test H0: in Z by 0. When Xi 0 by replacing are Poisson variables, this test is preferable to the largesample test of Section 9-2.5, which would use S 1n in the denominator, because it is designed just for the Poisson distribution. Suppose that the number of open circuits on a semiconductor wafer has a Poisson distribution. Test data for 500 wafers indicate a total of 1038 opens. Using 0.05, does this suggest that the mean number of open circuits per wafer exceeds 2.0

Use this result to test H0: 10 versus H1: 10 for the golf ball overall distance data in Exercise 6-25. 9-95. Continuation of Exercise 9-94. Using the results of the previous exercise, nd an approximately unbiased estimator of the 95 percentile 1.645 . From the fact that X and S are independent random variables, nd the standard error of . How would you estimate the standard error 9-96. Continuation of Exercises 9-94 and 9-95. Consider the golf ball overall distance data in Exercise 6-25. We wish to investigate a claim that the 95 percentile of overall distance does not exceed 285 yards. Construct a test statistic that can be used for testing the appropriate hypotheses. Apply this procedure to the data from Exercise 6-25. What are your conclusions 9-97. Let X1, X2, p , Xn be a sample from an exponential distribution with parameter . It can be shown that n 2 i 1 Xi has a chi-square distribution with 2n degrees of freedom. Use this fact to devise a test statistic and critical region for H0: 0 versus the three usual alternatives.

IMPORTANT TERMS AND CONCEPTS In the E-book, click on any term or concept below to go to that subject. Connection between hypothesis tests and con dence intervals Critical region for a test statistic Null hypothesis One- and two-sided alternative hypotheses Operating characteristic curves Power of the test P-value Reference distribution for a test statistic Sample size determination for hypothesis tests Signi cance level of a test Statistical hypotheses Statistical versus practical signi cance Test for goodness of t Test for homogeneity Test for independence Test statistic Type I and type II errors

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