Q6Single correct3 Marks30 Apr 2023

Consider two independent samples $X_1, X_2, \dots, X_{50} \sim \text{i.i.d. Normal}(\mu_1, 25)$ and $Y_1, Y_2, \dots, Y_{20} \sim \text{i.i.d. Normal}(\mu_2, 70)$. Let the null and alternative hypothesis be: $$H_0: \mu_1 = \mu_2$$ $$H_A: \mu_1 \neq \mu_2$$ Suppose $T = \overline{Y} - \overline{X}$, where $\overline{Y} = \frac{Y_1 + Y_2 + \dots + Y_{20}}{20}$ and $\overline{X} = \frac{X_1 + X_2 + \dots + X_{50}}{50}$. Consider a test that rejects $H_0$ if $|T| > c$ for some constant $c$. What is the size of the test in terms of 'c'?

A

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

B

$2F_Z\left(\frac{-c}{2} \right)$

C

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

D

$F_Z\left(\frac{c}{2} \right)$