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

Shape gives a product scale gives a sum

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

A. $\prod (i = 1.. n) X_{i} > C$ ✓
B. $\sum (i = 1.. n) X_{i} > C$
C. $\prod (i = 1.. n) X_{i} < C$
D. $\sum (i = 1.. n) X_{i} < C$

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.

See pricing

50 are analysed free — try those first.

Related counterexample: A UMP test exists for every testing problem

Shape gives a product scale gives a sum

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