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Part BCSIR NET December 2023direction-of-rejection

Direction of rejection

Let X1,…,X6X_{1}, \dots, X_{6} be a random sample from the gamma density f(x|λ)=(λ4/6)\lambda) = (\lambda^{4}/6) e−λxe^{-\lambda{}x} x3x^{3} for x>0,λ>0x > 0, \lambda > 0 unknown. Let T=∑XiT = \sum{}X_{i} and ψ\psi be the UMP test of size 0.05 for H0:λ=1H_{0}: \lambda = 1 against H1:λ>1H_{1}: \lambda > 1. With χν,α2\chi^{2}_{\nu,\alpha} the (1−α)−(1 - \alpha)-th quantile of χν2\chi^{2}_\nu, the test ψ\psi rejects H0H_{0} iff

  1. A.T ≥ ½ χ48,0.052\chi^{2}_{48, 0.05}
  2. B.T ≤ ½ χ48,0.952\chi^{2}_{48, 0.95}✓ Correct
  3. C.T ≥ ½ χ24,0.052\chi^{2}_{24, 0.05}
  4. D.T ≤ ½ χ24,0.952\chi^{2}_{24, 0.95}

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.

See pricing

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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