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Q30Section A (Section A)30 of 60

Power interval of a chi-square scale test

Let X1,X2,…,X10X_1,X_2,…,X_{10} be a random sample from a N(0,σ2)N(0,\sigma^2) distribution, where σ>0\sigma>0 is unknown. For testing H0:σ2≤1H_0:\sigma^2\le 1 against H1:σ2>1H_1:\sigma^2>1, a test of size α=0.05\alpha=0.05 rejects H0H_0 if and only if ∑i=110Xi2>18.307\sum_{i=1}^{10}X_i^2>18.307. Let β\beta be the power of this test, at σ2=2\sigma^2=2. Then β\beta lies in the interval (You may use χ10,0.052=18.307\chi^2_{10,0.05}=18.307, χ10,0.12=15.9872\chi^2_{10,0.1}=15.9872, χ10,0.252=12.5489\chi^2_{10,0.25}=12.5489, χ10,0.52=9.3418\chi^2_{10,0.5}=9.3418, χ10,0.752=6.7372\chi^2_{10,0.75}=6.7372, χ10,0.92=4.8652\chi^2_{10,0.9}=4.8652, χ10,0.952=3.9403\chi^2_{10,0.95}=3.9403, χ10,0.9752=3.247\chi^2_{10,0.975}=3.247)

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