George Washington University Interval Estimation & Pivotal Quantities Questions
Question Description
Let X be a random variable with continuous distribution function F(x; ?),and you wish to test H0 : ? = ?0.
(a) (5 points) Show that the test that rejects H0 if F(X; ?0) ? 0.025 orF(X; ?) ? 0.975 has level 0.05.
(b) (5 pts) Now apply the result in (a) to the setting where F(x; ?) is thedistribution function of a sample mean X¯ from iid N(µ, ?2), where?2is known. Show that the test in part (a) for testing H0 : ? = 0 isthe usual 2-tailed test rejecting H0 if Z ? ?1.96 or Z ? 1.96, whereZ = X/¯ (?/?n).
(c) (5 points) Invert the test in part (b) by finding the set of ?0 such thatthe test accepts the null hypothesis H0 : ? = ?0, and show that youget the usual confidence interval (X¯ ? 1.96?/?n, X¯ + 1.96?/?n).
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