statistics assignment

Statistical Inference II: J. Lee Assignment 4

Problem 1. (Exercise 12 in Chapter 9 of Rice) Let X1, . . . ,Xn be a random sample from an exponential distribution with the density function f(x|θ) = θe−θx. Derive a likelihood ratio test of H0 : θ = θ0 versus HA : θ 6= θ0, and show that the rejection region is of the form {Xe−θ0X ≤ c}.

Problem 2. (Exercise 2 in Chapter 9 of Rice) Which of the following hypotheses are simple, and which are composite?

(a) X follows a uniform distribution on [0,1].

(b) A die is unbiased.

(c) X follows a normal distribution with mean 0 and variance σ2 > 10.

(d) X follows a normal distribution with mean µ = 0.

Problem 3. (Exercise 18 in Chapter 9 of Rice) Let X1, . . . ,Xn be i.i.d. random variables from a double exponential distribution with density f(x) = 1

2 λe−λ|x|. Derive a likelihood ratio test of the hypothesis

H0 : λ = λ0 versus HA : λ = λ1, where λ0 and λ1 > λ0 are specified numbers. Is the test uniformly most powerful against the alternative H1 : λ > λ0?

Problem 4. Suppose under H0, X has the uniform distribution over (0,1) and under HA, X has the pdf of f(x) = 2x, 0 ≤ x ≤ 1.

(a) Find the most powerful lever α = 0.10 test of H0 vs. HA.

(b) Calculate the type II error probability of the test in (a).

Problem 5. Suppose that the EPA requires that a particular automobile model average at least 25 miles per gallon of gasoline. A sample of 16 cars are tested. For each of the cars, the mileage in miles per gallon is determined over a 500 km course. Assuming that the recorded mileages are normally distributed,

(a) Formulate the hypothesis testing problem appropriate for analyzing whether the manufacturer is in compliance and explain how one would test at a significance level of .05.

(b) Suppose that X = 24.8 and the sample standard deviation s = 1. What would you conclude about compliance if you were testing at a significance level of .10?