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?
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?
\[ \bar{X}_e = \frac{1}{25} \sum_{i=1}^{25} X_{2i}, \] \[ \bar{X}_o = \frac{1}{25} \sum_{i=1}^{25} X_{2i-1}, \] \[ S_e = \sqrt{\frac{1}{24} \sum_{i=1}^{25} (X_{2i} - \bar{X}_e)^2}, \] 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?
Updated On: Oct 1, 2024
- \( \frac{5\bar{X}_e}{S_e} \) has \( t_{24} \) distribution
- \( \frac{5(\bar{X}_e + \bar{X}_o)}{\sqrt{S_e^2 + S_o^2}} \) has \( t_{49} \) distribution
- \( \frac{49 S_o^2}{\sigma^2} \) has \( \chi_{49}^2 \) distribution
- \( \frac{S_o^2}{S_e^2} \) has \( F_{24,24} \) distribution
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The Correct Option is A, D
Solution and Explanation
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
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