Part CCSIR NET December 2023ump-mean-vs-median

Ump mean vs median

Let $X_{1}, \dots, X_{n}$ be a random sample from $N (μ, 1)$ . To test $H_{0} : μ = μ_{0}$ against $H_{1} : μ > μ_{0}$ , consider (A) reject iff $\overset{ˉ}{X}_{n} > c_{1}$ with $P_{μ_{0}} (X_{n} > c_{1}) = α$ , and (B) reject iff Median{ $X_{1}, \dots, X_{n}$ } $> c_{2}$ with $P_{μ_{0}} (M e d ian$ $> c_{2}) = α$ . Which of the following statements are true?

A.The test in (A) is the uniformly most powerful test of size $α$
B.The test in (B) is the uniformly most powerful test of size $α$
C. $P_{μ} (\overset{ˉ}{X}_{n} > c_{1}) \to 1$ as $n \to \infty$ for all $μ > μ_{0}$
D. $P_{μ_{0}} (M e d ian$ { $X_{1}, \dots, X_{n}$ } $> μ_{0}) = 1/2$

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p-values, size, power and error types say specific things. This was not one of them.

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Related counterexample: A UMP test exists for every testing problem

Ump mean vs median

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