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Part BCSIR NET December 2024randomise-at-the-boundary-to-hit-the-size

Randomise at the boundary to hit the size

Let X1,X2,…,X5X_{1}, X_{2}, \dots, X_{5} be a random sample of size 5 from an absolutely continuous distribution having median M. Let S denote the number of Xi′sX_{i}'s greater than 0. For testing H0:M=0H_{0} : M = 0 against H1:M>0H_{1} : M > 0, let φ(X)=1\varphi(X) = 1 if S>c,νS > c, \nu if S = c, and 0 if S < c be a test of size α=0.05\alpha = 0.05, where ν∈[0,1]\nu \in [0, 1] and c ∈ {−1, 0, …, 5} are fixed constants. Then c+νc + \nu equals

  1. A.11/25
  2. B.49/6
  3. C.103/25✓ Correct
  4. D.53/6

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: Any interval of the form [X̄ − (S/√n)t, ∞) with a 90% quantile t is a 90% confidence interval

More on this topic

The chapter behind this: Standard tests and confidence intervals — free to read

From Hypothesis Testing › Likelihood ratio and standard tests

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