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

The Correct Option is A, D

Step 1: Analyze statement (A). For \( \bar{X}_e = \frac{1}{25} \sum_{i=1}^{25} X_{2i} \), we know: \[ \frac{\bar{X}_e}{\sigma / \sqrt{25}} \sim N(0, 1). \] Also, \( S_e^2 \) is an unbiased estimator of \( \sigma^2 \) based on 24 degrees of freedom. Therefore: \[ \frac{\bar{X}_e}{S_e / \sqrt{25}} = \frac{5\bar{X}_e}{S_e} \sim t_{24}. \] Thus, \textbf{(A)} is correct. Step 2: Analyze statement (B). The random variables \( \bar{X}_e \) and \( \bar{X}_o \) are independent, and \( S_e^2 + S_o^2 \) is based on 48 degrees of freedom (not 49). Hence, \textbf{(B)} is incorrect because \( t_{49} \) is not the correct distribution. Step 3: Analyze statement (C). The statistic \( S_o^2 \) is an unbiased estimator of \( \sigma^2 \) based on 24 degrees of freedom. Therefore: \[ \frac{24S_o^2}{\sigma^2} \sim \chi_{24}^2, \] and not \( \chi_{49}^2 \). Hence, \textbf{(C)} is incorrect. Step 4: Analyze statement (D). The ratio \( \frac{S_o^2}{S_e^2} \) follows an \( F \)-distribution with \( 24, 24 \) degrees of freedom: \[ \frac{S_o^2}{S_e^2} \sim F_{24,24}. \] Thus, \textbf{(D)} is correct. Conclusion: The correct answers are: \[ \boxed{\text{(A)} \text{ and } \text{(D)}}. \]