Download Statistics Homework 3: Hypothesis Testing and Power Function and more Assignments Mathematical Statistics in PDF only on Docsity! Spring 2009 Statistics 131C Homework 3 Due : April 24 1. Problem 8.4.4 : Suppose that X1, . . . , Xn form a random sample from a normal distribu- tion for which the mean µ is unknown and the variance is 1, and it is desired to test the following hypotheses: H0 : 0.1 ≤ µ ≤ 0.2 against H1 : µ < 0.1 or µ > 0.2. Consider a test procedure δ such that the hypothesis H0 is rejected if either Xn ≤ c1 or Xn ≥ c2, and let π(µ|δ) denote the power function of δ. Suppose that the sample size is n = 25. Determine the values of the constants c1 and c2 such that π(0.1|δ) = π(0.2|δ) = 0.07. 2. Problem 8.4.5 : Consider again the conditions of Problem 8.4.4, and suppose also that n = 25. Determine the values of the constants c1 and c2 such that π(0.1|δ) = 0.02 and π(0.2|δ) = 0.05. 3. Problem 8.4.10 : Suppose that X1, . . . , Xn form a random sample from a uniform distribu- tion on the interval [0, θ]. Suppose now that it is desired to test the following hypotheses: H0 : θ = 3 against H1 : θ 6= 3. Consider a test procedure δ such that the hypothesis H0 is rejected if either max{X1, . . . , Xn} ≤ c1 or max{X1, . . . , Xn} ≥ c2, and let π(θ|δ) denote the power function of δ. Determine the values of the constants c1 and c2 such that π(3|δ) = 0.05 and δ is unbiased. 4. Problem 8.5.3 : The manufacturer of a certain type of automobile claims that under typical urban driving conditions the automobile will travel on average at least 20 miles per gallon of gasoline. The owner of this type of automobile notes the mileages that she has obtained in her own urban driving when she fills her automobile’s tank with gasoline on nine different occasions. She finds that the results, in miles per gallon, are as follows: 15.6, 18.6, 18.3, 20.1, 21.5, 18.4, 19.1, 20.4, 19.0. Test the manufacturer’s claim by carrying out a test at the level of significance α0 = 0.05. List carefully the assumptions you must make. 5. Problem 8.5.6 : Suppose that the variables X1, . . . , Xn form a random sample from a normal distribution for which both the mean µ and variance σ2 are unknown, and a t test at a given level of significance α0 is to be carried out to test the following hypotheses: H0 : µ ≤ µ0 against H1 : µ > µ0. 1
