Question

Let \( x_1, x_2, x_3, x_4 \) be the observed values from a random sample drawn from a \( N(\mu, \sigma^2) \) distribution, where \( \mu \in \mathbb{R} \) and \( \sigma \in (0, \infty) \) are unknown parameters. Let \( \bar{x} \) and \( s = \sqrt{\frac{1}{3} \sum_{i=1}^{4} (x_i - \bar{x})^2} \) be the observed be the observed sample mean sample standard deviation,repectively. For testing the hypotheses \( H_0: \mu = 0 \) against \( H_1: \mu \neq 0 \), the likelihood ratio test of size \( \alpha = 0.05 \) rejects \( H_0 \) if and only if \[\frac{|\bar{x}|}{s} > k.\] Then the value of \( k \) is given by:

• A. \( \frac{1}{2} t_{3,0.025} \)
• B. \( t_{3,0.025} \)
• C. \( 2 t_{3,0.05} \)
• D. \( \frac{1}{2} t_{3,0.05} \)

Answer 1

The Correct Option is A

1. Likelihood Ratio Test Setup: - The test statistic is \( \frac{|\bar{x}|}{s} \), which follows a \( t \)-distribution with \( n-1 = 4-1 = 3 \) degrees of freedom under the null hypothesis \( H_0 \). 
2.Significance Level and Critical Value: - The test is of size \( \alpha = 0.05 \), which means the rejection region corresponds to the upper 2.5\% of the \( t \)-distribution in a two-tailed test. - The critical value is given by \( t_{3, 0.025} \) for 3 degrees of freedom. 
3. Scaling of the Test Statistic:}- For the test statistic \( \frac{|\bar{x}|}{s} \) to match the critical value \( k \), the relationship \( k = \frac{1}{2} t_{3, 0.025} \) holds true.