Question

Suppose ?1 has normal distribution with expectation 0 and variance ? 2 , ?2 has normal distribution with expectation 0 and variance 2? 2 . ?0 : ? 2 = ?, ?1 : ? 2 > ?, where ? > 0. Recall that the one sided ? 2 test for the null hypothesis ? ∼ ? 2 (?) is ? ≥ ? −1 ?2(?) (1 − ?), here ? is the c.d.f. of ? 2 (?), and ? is the significance level.

1. Find a numbers ? and ? such that under ?0, ??1^2 + ??2^2 ∼ ? 2 (2).

2. Use the one-sided ? 2 test described above to write down a statistical test of ?0 against ?1.

3. Find the one-sided confidence interval using the test you found above.

Answer 1

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