Q28Multiple correct3 Marks24 Dec 2023

Suppose $X_1, X_2, \dots, X_n \sim \text{iid Normal}(\mu, \sigma^2)$ with sample mean, $\bar{X} = \frac{\sum_{i=1}^n X_i}{n}$ and sample variance, $S^2 = \frac{\sum_{i=1}^n (X_i - \bar{X})^2}{n - 1}$. Choose the correct option(s) from the following.

A

$\frac{\bar{X} - \mu}{(S/\sqrt{n})} \sim N(0, 1)$

B

$\frac{(n-1)S^2}{\sigma^2} \sim \chi^2_{n-1}$

C

$\frac{\bar{X} - \mu}{(\sigma/\sqrt{n})} \sim t_{n-1}$

D

$\frac{\bar{X} - \mu}{(S/\sqrt{n})} \sim t_{n-1}$