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.