Question

Let $X_1, X_2, \ldots, X_{50}$ be a random sample from a $N(0, \sigma^2)$ distribution, where $\sigma > 0$. Define
$\bar{X}_e = \frac{1}{25} \sum_{i=1}^{25} X_{2i},$ $\bar{X}_o = \frac{1}{25} \sum_{i=1}^{25} X_{2i-1},$ $S_e = \sqrt{\frac{1}{24} \sum_{i=1}^{25} (X_{2i} - \bar{X}_e)^2},$ and $S_o = \sqrt{\frac{1}{24} \sum_{i=1}^{25} (X_{2i-1} - \bar{X}_o)^2}.$
Then which of the following statements is/are correct?

• A. $\frac{5\bar{X}_e}{S_e}$ has $t_{24}$ distribution
• B. $\frac{5(\bar{X}_e + \bar{X}_o)}{\sqrt{S_e^2 + S_o^2}}$ has $t_{49}$ distribution
• C. $\frac{49 S_o^2}{\sigma^2}$ has $\chi_{49}^2$ distribution
• D. $\frac{S_o^2}{S_e^2}$ has $F_{24,24}$ distribution

Answer 1

Answer: (A), (D)

$\frac{5\bar{X}_e}{S_e}$ has $t_{24}$ distribution

Answer 2

The correct option is (A):$\frac{5\bar{X}_e}{S_e}$ has $t_{24}$ distribution,(D): $\frac{S_o^2}{S_e^2}$ has $F_{24,24}$ distribution