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

Ump mean vs median

Let $X_{1}, \dots, X_{n}$ be a random sample from $N(\mu, 1)$ . To test $H_{0}: \mu = \mu_{0}$ against $H_{1}: \mu > \mu_{0}$ , consider (A) reject iff ${\bar{X}}_{n} > c_{1}$ with $P_{\mu_{0}}(X_{n} > c_{1}) = \alpha$ , and (B) reject iff Median{ $X_{1}, \dots, X_{n}$ } $> c_{2}$ with $P_{\mu_{0}}(Median$ $> c_{2}) = \alpha$ . Which of the following statements are true?

A.The test in (A) is the uniformly most powerful test of size $\alpha$ ✓ Correct
B.The test in (B) is the uniformly most powerful test of size $\alpha$
C. $P_\mu({\bar{X}}_{n} > c_{1}) \to 1$ as $n \to \infty$ for all $\mu > \mu_{0}$ ✓ Correct
D. $P_{\mu_{0}}(Median$ { $X_{1}, \dots, X_{n}$ } $> \mu_{0}) = 1/2$ ✓ Correct

You have the answer. Trap Analysis is why the other three were written.

Not a worked solution repeated four times — the specific reasoning error each wrong option was built to reward, and why this one tests what the inference means.

See pricing

50 are analysed free — try those first.

The trap it tests

What the inference means

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

Drill statements like this

Related counterexample: A UMP test exists for every testing problem

Ump mean vs median

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