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Part BCSIR NET December 2024the-reversed-size-is-one-minus-the-power

The reversed size is one minus the power

Suppose φ\varphi is a most powerful test of size 0.05 for testing a simple null hypothesis H0H_{0} against a simple alternative hypothesis H1H_{1}. If the power of the test is 0.4, then which of the following is true?

  1. A.(1−φ)(1 - \varphi) is a most powerful test at level 0.6 for testing H1H_{1} against H0H_{0}.✓ Correct
  2. B.(1−φ)(1 - \varphi) is a most powerful test at level 0.4 for testing H1H_{1} against H0H_{0}.
  3. C.(1−φ)(1 - \varphi) is a most powerful test at level 0.05 for testing H1H_{1} against H0H_{0}.
  4. D.(1−φ)(1 - \varphi) is NOT a most powerful test for testing H1H_{1} against H0H_{0} at any level.

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.

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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

More on this topic

The chapter behind this: Neyman–Pearson and UMP tests — free to read

From Hypothesis Testing › Neyman–Pearson lemma and UMP tests

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