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Part BCSIR NET December 2024shape-gives-a-product-scale-gives-a-sum

Shape gives a product scale gives a sum

Let X1,X2,…,Xn(n≥2)X_{1}, X_{2}, \dots, X_{n} (n \ge 2) be a random sample from a gamma distribution with shape parameter α>0\alpha > 0 and scale parameter β=1\beta = 1. For a suitable constant C, the rejection region of the most powerful test for testing H0:α=1H_{0} : \alpha = 1 against H1:α=2H_{1} : \alpha = 2 is of the form

  1. A.∏(i=1..n)Xi>C\prod(i=1..n) X_{i} > C✓ Correct
  2. B.∑(i=1..n)Xi>C\sum(i=1..n) X_{i} > C
  3. C.∏(i=1..n)Xi<C\prod(i=1..n) X_{i} < C
  4. D.∑(i=1..n)Xi<C\sum(i=1..n) X_{i} < C

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.

See pricing

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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