Question

Let \( X_1, X_2, \ldots, X_{10} \) be a random sample from a \( N(0, \sigma^2) \) distribution, where \( \sigma > 0 \) is unknown. For testing \( H_0: \sigma^2 \leq 1 \) against \( H_1: \sigma^2 > 1 \), a test of size \( \alpha = 0.05 \) rejects \( H_0 \) if and only if \( \sum_{i=1}^{10} X_i^2 > 18.307 \). Let \( \beta \) be the power of this test, at \( \sigma^2 = 2 \). Then \( \beta \) lies in the interval
(You may use \( \chi^2_{10,0.05} = 18.307 \), \( \chi^2_{10,0.1} = 15.9872 \), \( \chi^2_{10,0.25} = 12.5489 \), \( \chi^2_{10,0.5} = 9.3418 \), \( \chi^2_{10,0.75} = 6.7372 \), \( \chi^2_{10,0.9} = 4.8652 \), \( \chi^2_{10,0.95} = 3.9403 \), \( \chi^2_{10,0.975} = 3.247 \))

• A. (0.50, 0.75)
• B. (0.75, 0.90)
• C. (0.90, 0.95)
• D. (0.95, 0.975)

Answer 1

The Correct Option is A

1. Test Statistic: Under \( H_0 \), \( \sum_{i=1}^{10} X_i^2 \sim \chi^2_{10} \). Under \( H_1 \) with \( \sigma^2 = 2 \), \( \sum_{i=1}^{10} X_i^2 \sim \frac{\chi^2_{10}}{2} \). 

2. Power of the Test: - The rejection region is \( \sum_{i=1}^{10} X_i^2 > 18.307 \). - Evaluate the probability under \( H_1 \) with \( \sigma^2 = 2 \) using the chi-squared distribution scaled by \( \frac{1}{2} \). 

3. Interval for Power \( \beta \): Comparing the cumulative probabilities for the given critical values, \( \beta \) lies in the interval \( (0.50, 0.75) \)