Question:
The Neyman-Pearson Lemma can be used to test the hypothesis :
The Neyman-Pearson Lemma can be used to test the hypothesis :
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The Neyman-Pearson Lemma needs a simple hypothesis, meaning every parameter fixed at one exact value, not a range or inequality.
Updated On: Jul 4, 2026
- \(H_0 : \mu = 4, \sigma = 1\)
- \(H_0 : \mu = 4, \sigma > 1\)
- \(H_0 : \mu = 0, \sigma \neq 1\)
- All of these
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The Correct Option is A
Solution and Explanation
Step 1: The Neyman-Pearson Lemma gives the most powerful test only for testing a simple null hypothesis against a simple alternative hypothesis, where "simple" means every parameter of the distribution is fixed at one specific numerical value, leaving no free or ranging parameter.
Step 2: Check each option against this requirement. Option (A), \(H_0 : \mu = 4, \sigma = 1\), fixes both the mean and the standard deviation at single values, so the distribution under \(H_0\) is completely and uniquely specified; this is a simple hypothesis.
Step 3: Option (B), \(H_0 : \mu = 4, \sigma > 1\), does not fix \(\sigma\) to one value but allows a whole range of values greater than 1, so it is a composite hypothesis. Option (C), \(H_0 : \mu = 0, \sigma \neq 1\), similarly allows \(\sigma\) to take any value except 1, again a range of values, so it is also composite. The classical Neyman-Pearson Lemma does not directly apply to composite hypotheses like (B) and (C).
Final Answer: Only option (A) is a simple hypothesis testable directly by the Neyman-Pearson Lemma. \[\boxed{H_0 : \mu = 4, \sigma = 1}\]
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