Part CCSIR NET December 2023comparing-test-statistics

Comparing test statistics

Let $X_{1}, \dots, X_{25}$ be i.i.d. Bernoulli(p), X̄ their mean, $T_{1} = 5({\bar{X}} - 0.5)/\sqrt{{\bar{X}}(1 - {\bar{X}})}$ for 0 < X̄ < 1 (±5 at the endpoints) and $T_{2} = 10({\bar{X}} - 0.5)$ . For $H_{0}: p = 0.5$ vs $H_{1}: p > 0.5$ , test $\psi_{i}$ rejects iff $T_{i} > 2$ . If the observed X̄ ∈ (0.5, 0.75), which of the following statements are true?

A.If $\psi_{1}$ rejects $H_{0}$ , then $\psi_{2}$ also rejects $H_{0}$
B.If $\psi_{1}$ does not reject $H_{0}$ , then $\psi_{2}$ also does not reject $H_{0}$ ✓ Correct
C.If $\psi_{2}$ rejects $H_{0}$ , then $\psi_{1}$ also rejects $H_{0}$ ✓ Correct
D.If $\psi_{2}$ does not reject $H_{0}$ , then $\psi_{1}$ also does not reject $H_{0}$

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

Comparing test statistics

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