Question

Let X₁, X₂ , … , X_n (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 c_α. 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

The Correct Option is A, B

To solve this problem, we need to understand the concept of hypothesis testing and the definitions of test size and power.

Conceptual Understanding

• The size of a test, denoted by \( \alpha \), is the probability of rejecting the null hypothesis \( H_0 \) when it is true (Type I error).
• The power of a test, denoted by \( \beta \), is the probability of rejecting the null hypothesis \( H_0 \) when a specific alternative hypothesis \( H_1 \) is true (1 minus the probability of Type II error).

Given

We have a random sample from a normal distribution \( N(\mu, 1) \), and we are testing

• \( H_0: \mu = 0 \) against \( H_1: \mu = \delta \), where \( \delta > 0 \).

The test rejects \( H_0 \) if and only if

\(\frac{1}{n}\sum^n_{i=1}X_i > c_{\alpha}\)

where \( c_{\alpha} \) is a constant determined by the test size \( \alpha \).

Analysis of Options

1. For a fixed value of \( \delta \), \( \beta \) increases as \( \alpha \) increases.
   • Increasing \( \alpha \) leads to a higher chance of rejecting \( H_0 \) when true, hence increasing the chance when \( H_1 \) is true, thereby increasing power.
2. For a fixed value of \( \alpha \), \( \beta \) increases as \( \delta \) increases.
   • A larger \( \delta \) increases the mean under \( H_1 \), making it more distinguishable from the mean under \( H_0 \), thereby increasing power.
3. For a fixed value of \( \delta \), \( \beta \) decreases as \( \alpha \) increases.
   • This is incorrect as explained in option 1, power actually increases with \( \alpha \).
4. For a fixed value of \( \alpha \), \( \beta \) decreases as \( \delta \) increases.
   • This is incorrect as explained in option 2, power increases with a larger difference defined by \( \delta \).

Conclusion

Based on the analysis, the correct statements are:

• For a fixed value of \( \delta \), \( \beta \) increases as \( \alpha \) increases.
• For a fixed value of \( \alpha \), \( \beta \) increases as \( \delta \) increases.