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Part CCSIR NET December 2024the-upper-tail-costs-no-size

The upper tail costs no size

Let X1,X2,X3X_{1}, X_{2}, X_{3} be a random sample from Uniform[0,θ][0, \theta] distribution, θ>0\theta > 0. Consider the likelihood ratio test of size 0.001 for testing H0:θ=3H_{0} : \theta = 3 against H1:θ≠3H_{1} : \theta \ne 3. Then, which of the following statements are true?

  1. A.If max{X1,X2,X3X_{1}, X_{2}, X_{3}} is 3.1, then H0H_{0} is rejected.✓ Correct
  2. B.If max{X1,X2,X3X_{1}, X_{2}, X_{3}} is 1.3, then H0H_{0} is rejected.
  3. C.If max{X1,X2,X3X_{1}, X_{2}, X_{3}} is 0.1, then H0H_{0} is rejected.✓ Correct
  4. D.The power of the test at θ=0.3\theta = 0.3 is 1.✓ 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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