Examples and Problems in Mathematical Statistics

PART III: PROBLEMS

Section 4.1

4.1.1 Consider Example 4.1. It was suggested to apply the test statistic (X) = I{X ≤ 18}. What is the power of the test if (i) θ = .6; (ii) θ = .5; (iii) θ = .4? [Hint: Compute the power exactly by applying the proper binomial distributions.]

4.1.2 Consider the testing problem of Example 4.1 but assume that the number of trials is n = 100.

(i) Apply the normal approximation to the binomial to develop a large sample test of the hypothesis H0: θ ≥ .75 against the alternative H1: θ < .75.

(ii) Apply the normal approximation to determine the power of the large sample test when θ = .5.

(iii) Determine the sample size n according to the large sample formulae so that the power of the test, when θ = .6, will not be smaller than 0.9, while the size of the test will not exceed α = .05.

4.1.3 Suppose that X has a Poisson distribution with mean λ. Consider the hypotheses H0: λ = 20 against H1: λ ≠ 20.

(i) Apply the normal approximation to the Poisson to develop a test of H0 against H1 at level of significance α = .05.

(ii) Approximate the power function and determine its value when λ = 25.

Section 4.2

4.2.1 Let X1, …, Xn be i. i. d. random variables having a common negative–binomial distribution NB(, ν), where ν is known.

(i) Apply the Neyman–Pearson Lemma ...