Question:
Let \(X_1,X_2,\ldots,X_n\) be a random sample of size \(n\ (n\geq2)\) from a \(N(\mu,1)\) distribution, where \(\mu\in\mathbb{R}\) is an unknown parameter. Consider the problem of testing \(H_0:\mu=5\) against \(H_1:\mu\neq5\). Which one of the following statements is true?
Let \(X_1,X_2,\ldots,X_n\) be a random sample of size \(n\ (n\geq2)\) from a \(N(\mu,1)\) distribution, where \(\mu\in\mathbb{R}\) is an unknown parameter. Consider the problem of testing \(H_0:\mu=5\) against \(H_1:\mu\neq5\). Which one of the following statements is true?
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For a two-sided test, the power function is lowest at the null value and increases as the true parameter moves away from the null value.
Updated On: Jun 4, 2026
- The power function of the likelihood ratio test at level \(0.05\) has a local maximum at \(\mu=5\)
- The power function of the likelihood ratio test at level \(0.05\) is decreasing on \((5,\infty)\)
- The power function of the likelihood ratio test at level \(0.05\) is decreasing on \((-\infty,5)\)
- The power function of the likelihood ratio test at level \(0.05\) is increasing on \((0,5)\)
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The Correct Option is C
Solution and Explanation
Step 1: Understand the testing problem.
The hypotheses are
\[ H_0:\mu=5 \] against
\[ H_1:\mu\neq5 \]
This is a two-sided test for the mean of a normal distribution with known variance.
Step 2: Identify the nature of the likelihood ratio test.
For a two-sided likelihood ratio test, we reject \(H_0\) when the sample mean is sufficiently far away from \(5\).
Thus, the rejection region is of the form
\[ |\bar{X}-5|>c \]
for some critical value \(c>0\).
Step 3: Understand the power function.
The power function gives the probability of rejecting \(H_0\) when the true value of the parameter is \(\mu\).
At \(\mu=5\), the probability of rejection is equal to the level of the test, which is
\[ 0.05 \]
As \(\mu\) moves away from \(5\), the probability of rejection increases.
Step 4: Analyze behaviour on \((5,\infty)\).
For
\[ \mu>5, \]
as \(\mu\) increases farther away from \(5\), the sample mean is more likely to fall in the rejection region.
Therefore, the power function increases on
\[ (5,\infty) \]
Hence, option (B) is incorrect.
Step 5: Analyze behaviour on \((-\infty,5)\).
For
\[ \mu<5, \] the farther \(\mu\) is from \(5\), the larger the power.
As \(\mu\) increases from \(-\infty\) toward \(5\), it moves closer to the null value.
Therefore, the power function decreases on
\[ (-\infty,5) \]
Hence, option (C) is correct.
Step 6: Final conclusion.
The power function is minimum near the null value \(\mu=5\) and increases as \(\mu\) moves away from \(5\).
Therefore, the correct statement is
\[ \boxed{(C)} \]
The hypotheses are
\[ H_0:\mu=5 \] against
\[ H_1:\mu\neq5 \]
This is a two-sided test for the mean of a normal distribution with known variance.
Step 2: Identify the nature of the likelihood ratio test.
For a two-sided likelihood ratio test, we reject \(H_0\) when the sample mean is sufficiently far away from \(5\).
Thus, the rejection region is of the form
\[ |\bar{X}-5|>c \]
for some critical value \(c>0\).
Step 3: Understand the power function.
The power function gives the probability of rejecting \(H_0\) when the true value of the parameter is \(\mu\).
At \(\mu=5\), the probability of rejection is equal to the level of the test, which is
\[ 0.05 \]
As \(\mu\) moves away from \(5\), the probability of rejection increases.
Step 4: Analyze behaviour on \((5,\infty)\).
For
\[ \mu>5, \]
as \(\mu\) increases farther away from \(5\), the sample mean is more likely to fall in the rejection region.
Therefore, the power function increases on
\[ (5,\infty) \]
Hence, option (B) is incorrect.
Step 5: Analyze behaviour on \((-\infty,5)\).
For
\[ \mu<5, \] the farther \(\mu\) is from \(5\), the larger the power.
As \(\mu\) increases from \(-\infty\) toward \(5\), it moves closer to the null value.
Therefore, the power function decreases on
\[ (-\infty,5) \]
Hence, option (C) is correct.
Step 6: Final conclusion.
The power function is minimum near the null value \(\mu=5\) and increases as \(\mu\) moves away from \(5\).
Therefore, the correct statement is
\[ \boxed{(C)} \]
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