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?

• A. The power function of the likelihood ratio test at level \(0.05\) has a local maximum at \(\mu=5\)
• B. The power function of the likelihood ratio test at level \(0.05\) is decreasing on \((5,\infty)\)
• C. The power function of the likelihood ratio test at level \(0.05\) is decreasing on \((-\infty,5)\)
• D. The power function of the likelihood ratio test at level \(0.05\) is increasing on \((0,5)\)

Hint:

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.

Answer 1

The Correct Option is C

Step 1: Picture the test.
For $H_0:\mu=5$ against $\mu\neq5$ with known variance, we reject when $|\bar X-5|$ is large.

Step 2: Think about the power curve.
The power is the chance of rejecting at a true mean $\mu$. At $\mu=5$ it equals the level $0.05$, and it climbs as $\mu$ drifts away from $5$ on either side.

Step 3: Behaviour right of $5$.
For $\mu>5$, moving further out only makes rejection more likely, so the power rises on $(5,\infty)$. That kills option (B).

Step 4: Behaviour left of $5$.
For $\mu<5$, as $\mu$ increases toward $5$ it gets closer to the null value, so the power drops. Hence the power is decreasing on $(-\infty,5)$, which is option (C). It also rules out (D), since on $(0,5)$ the power falls.

Step 5: Conclude.
\[ \boxed{(C)} \]