confidence intervals

P5 Consider the Normal model M(u, o?) with / E R and o? > 0. (a) Using a suitable pivotal quantity, show how to construct a confidence interval for o' with confidence level 95%. Give examples of confidence intervals with confidence level 95% of the form (i) (L(X, U(X)), (ii) (L(X), co), and (iii) (-oo, U(X)), where X = (X1, .. ., Xn). (b) Consider the one-sided confidence interval for of of the form (L(X), co) with confidence level 95% that you constructed in part (a). For a suitably chosen set of undesirable parameters K-2, show that (L(X ), co) is unbiased. (c) Consider the one-sided confidence interval for o' of the form (L(X), co) with confidence level 95% that you constructed in part (a). Find the family of tests {vog,03 > 0} which corresponds to this confidence interval. For each of the tests 12, state the null and alternative hypothesis and identify the level of the test. (d) Consider the two-sided confidence interval for o of the form (L(X), U(X)) with confidence level 95% that you constructed in part (a). Find the family of tests (12,08 > 0} which corresponds to this confidence interval. For each of the tests 1 2, state the null and alternative hypothesis and identify the level of the test