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

Answer: (A)

$\frac{1}{2} t_{3,0.025}$

Answer 2

The correct option is (A): $\frac{1}{2} t_{3,0.025}$