Question

Let X1, X2 , … , Xn (n > 1) be a random sample from a N(μ, 1) distribution, where μ ∈ $\R$ is unknown. Let 0 < α < 1. To test the hypothesis H0 : μ = 0 against H1 : μ = δ, where δ > 0 is a constant, let β denote the power of the size α test that rejects H0 if and only if $\frac{1}{n}\sum^n_{i=1}X_i > c_{\alpha}$ , for some constant cα. Then which of the following statements is/are true ?

• A. For a fixed value of 𝛿, 𝛽 increases as 𝛼 increases
• B. For a fixed value of 𝛼, 𝛽 increases as 𝛿 increases
• C. For a fixed value of 𝛿, 𝛽 decreases as 𝛼 increases
• D. For a fixed value of 𝛼, 𝛽 decreases as 𝛿 increases

Answer 1

Answer: (A), (B)

For a fixed value of 𝛿, 𝛽 increases as 𝛼 increases

Answer 2

The correct option is (A) : For a fixed value of 𝛿, 𝛽 increases as 𝛼 increases and (B) : For a fixed value of 𝛼, 𝛽 increases as 𝛿 increases.