Q1Multiple correct3 Marks10 May 2026

Let $X, X_1, \dots, X_{16}$ be i.i.d. samples from $N(\mu_X, 8)$ and $Y, Y_1, \dots, Y_{36}$ be i.i.d. samples from $N(\mu_Y, 18)$. The observed sample means are $\bar{X} = 41$ and $\bar{Y} = 40$. We wish to test the hypotheses: $H_0: \mu_X = \mu_Y$ against $H_1: \mu_X \ne \mu_Y$. Consider a test that rejects $H_0$ if $|\bar{X} - \bar{Y}| > c$. Let $\alpha$ denote the significance level of the test and $F_Z(\cdot)$ denote the cumulative distribution function (CDF) of the standard normal distribution. Express the critical value $c$ in terms of $\alpha$ and $F_Z^{-1}(\cdot)$. Select all the option(s) that apply.

A

$c = F_Z^{-1}(\alpha)$

B

$c = F_Z^{-1}\left(1 - \frac{\alpha}{2}\right)$

C

$c = -F_Z^{-1}(1 - \alpha)$

D

$c = -F_Z^{-1}\left(\frac{\alpha}{2}\right)$