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 \( \chi^2_{10,0.05} = 18.307 \), \( \chi^2_{10,0.1} = 15.9872 \), \( \chi^2_{10,0.25} = 12.5489 \), \( \chi^2_{10,0.5} = 9.3418 \), \( \chi^2_{10,0.75} = 6.7372 \), \( \chi^2_{10,0.9} = 4.8652 \), \( \chi^2_{10,0.95} = 3.9403 \), \( \chi^2_{10,0.975} = 3.247 \))

The correct option is (A): (0.50, 0.75)

Let 𝑋1,𝑋2, … , 𝑋10 be a random sample from a 𝑁(0, 𝜎 2 ) distribution, where σ > 0 is unknown. For testing 𝐻0: 𝜎 2 ≤ 1 against 𝐻1: 𝜎 2 > 1, a test of size 𝛼 = 0.05 rejects 𝐻0 if and only if ∑ 𝑋𝑖 2 > 18.307 10 𝑖=1 . Let 𝛽 be the power of this test, at 𝜎 2 = 2. Then 𝛽 lies in the interval (You may use 𝜒10,0.05 2 = 18.307, 𝜒10,0.1 2 = 15.9872, 𝜒10,0.25 2 = 12.5489, 𝜒10,0.5 2 = 9.3418, 𝜒10,0.75 2 = 6.7372, 𝜒10,0.9 2 = 4.8652, 𝜒10,0.95 2 = 3.9403, 𝜒10,0.975 2 = 3.247 )