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Part CCSIR NET June 2025the-shifted-up-sample-is-the-x-one-so-large-mu-drags-the-y-ranks-down

The shifted up sample is the x one so large mu drags the y ranks down

Let (X1,X2,…,X7)(X_{1}, X_{2}, \dots, X_{7}) and (Y1,Y2,…,Y9)(Y_{1}, Y_{2}, \dots, Y_{9}) be two independent random samples from the continuous distribution functions F(x−μ)F(x - \mu) and F(x−θ)F(x - \theta), respectively, where F,μF, \mu and θ\theta are all unknown. Further, let μ\mu be the unique median of F(x−μ)F(x - \mu) and θ\theta be the unique median of F(x−θ)F(x - \theta). Let RiR_{i} be the rank of YiY_{i} in the combined sample, i = 1, 2, …, 9. For testing H0:μ=θH_{0} : \mu = \theta against H1:μ>θH_{1} : \mu > \theta, the test statistic T=∑iT = \sum_{i}₌19Ri_{1}^{9} R_{i} is proposed. Then, which of the following statements are true?

  1. A.The maximum possible value of T is 115
  2. B.Right-tailed test based on T is appropriate for testing H0H_{0} against H1H_{1}
  3. C.Under H0,E(T)=76H_{0}, E(T) = 76
  4. D.Under H0,P(R1=1,R9=16)=1/240H_{0}, P(R_{1} = 1, R_{9} = 16) = 1/240✓ Correct

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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