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, $T_{1} = 5 (\overset{ˉ}{X} - 0.5) / \overset{ˉ}{X} (1 - \overset{ˉ}{X})$ for 0 < X̄ < 1 (±5 at the endpoints) and $T_{2} = 10 (\overset{ˉ}{X} - 0.5)$ . For $H_{0} : p = 0.5$ vs $H_{1} : p > 0.5$ , test $ψ_{i}$ rejects iff $T_{i} > 2$ . If the observed X̄ ∈ (0.5, 0.75), which of the following statements are true?

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

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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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