Q3Multiple correct3 Marks22 Dec 2024

Suppose $X_1, X_2, \dots, X_{16} \sim \text{i.i.d. Normal}(\mu_1, 9)$ and $Y_1, Y_2, \dots, Y_8 \sim \text{i.i.d. Normal}(\mu_2, 25)$ are independent samples. The sample means of these samples are denoted as $\bar{X}$ and $\bar{Y}$ respectively. Let the null and alternative hypothesis be $H_0: \mu_1 = \mu_2$ and $H_A: \mu_1 \neq \mu_2$. Choose the correct option(s) from the following.

A

$\bar{X} - \bar{Y} \sim N\left(\mu_1 - \mu_2, \frac{9}{16} - \frac{25}{8}\right)$

B

Test Statistic: $T = \bar{X} - \bar{Y}$, Hypothesis test: Z-test.

C

Test Statistic: $T = \bar{X} - \bar{Y}$, Hypothesis test: t-test.

D

$\bar{X} - \bar{Y} \sim N\left(\mu_1 - \mu_2, \frac{9}{16} + \frac{25}{8}\right)$