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Part CCSIR NET December 2024one-value-of-the-statistic-arises-twice

One value of the statistic arises twice

Let X1,X2,X3X_{1}, X_{2}, X_{3} be a random sample of size 3 from an absolutely continuous distribution that is symmetric about 0. For i = 1, 2, 3, let RiR_{i} denote the rank of |XiX_{i}| among |X1X_{1}|, |X2X_{2}| and |X3X_{3}|. If T⁺ =∑(i:Xi>0)Ri= \sum(i : X_{i}>0) R_{i} is the Wilcoxon signed-rank statistic, then which of the following statements are true?

  1. A.P(T⁺ = 3) = 1/4✓ Correct
  2. B.Var(T⁺) = 7/2✓ Correct
  3. C.P(T⁺ > 3) = 5/8
  4. D.P(T⁺ > 4) = 1/8

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: 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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