1. Let Y_MAX be the sample minimum of a random sample of size 5 from a uniform distribution on the interval [0, b].

Consider the test with decision rule:

Reject b = 20 in favor of b > 20 when Y_MAX ≥ c.

a) Find the value of c for a 5% test.

b) Find the power of the test in part a when b = 25.

c) Assume the following data are the value of a random sample from the Y distribution.

11.74, 14.59, 17.71, 19.42, 19.71

Would the null hypothesis be accepted or rejected in this case?

2) Let X₁, X₂, X₃, X₄ be a random sample from a Poisson distribution with parameter λ and let Y = X₁ + X₂ + X₃ + X₄. You decide to test λ = 1.60 versus λ < 1.60 using following decision rule:

Reject λ = 1.60 in favor of λ < 1.60 when Y ≤ 2.

a) Find the Type I error, α.

b) Find the Type II error, β, when λ = 1.70.