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Part CCSIR NET December 2024the-stated-theta-values-are-built-to-give-round-probabilities

The stated theta values are built to give round probabilities

The life (in hours) of an electrical component is exponentially distributed with mean θ>0\theta > 0. For testing H0:θ=5/(ln⁡(10)−ln⁡(9))H_{0} : \theta = 5/(\ln(10) - \ln(9)) against H1:θ=5/ln⁡(2)H_{1} : \theta = 5/\ln(2), three such components are chosen at random. Suppose that we reject the null hypothesis H0H_{0} if and only if two or more of these three components survive for less than five hours. Then, which of the following statements are true?

  1. A.The size of the test is 0.01.
  2. B.The power of the test is 0.5.✓ Correct
  3. C.The test is unbiased at level α=0.05\alpha = 0.05.✓ Correct
  4. D.If all of these three components survive for less than five hours, then the p-value of the test is 0.001.✓ 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 execution slip.

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50 are analysed free — try those first.

The trap it tests

Execution slip

The idea was right. The computation was not.

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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