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

Answer: (A)

(0.50, 0.75)

Answer 2

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