Question

Let \(X_1, X_2, \ldots, X_{10}\) be a random sample from the \(N(3,4)\) distribution, and let \(Y_1, Y_2, \ldots, Y_{15}\) be a random sample from the \(N(-3,6)\) distribution. Assume that the two samples are drawn independently. Define:
\[ \bar{X} = \frac{1}{10} \sum_{i=1}^{10} X_i \]
\[ \bar{Y} = \frac{1}{15} \sum_{j=1}^{15} Y_j \]
\[ S = \sqrt{\frac{1}{9} \sum_{i=1}^{10} (X_i - \bar{X})^2} \]
Then the distribution of \( U = \frac{\sqrt{5}(\bar{X} + \bar{Y})}{S} \) is:

• A. \(N\left(0, \frac{4}{5}\right)\)
• B. \(\chi^2_9\)
• C. \(t_9\)
• D. \(t_{23}\)

Answer 1

The Correct Option is C

1. Mean and Variance of \(\bar{X}\) and \(\bar{Y}\): \[ \bar{X} \sim N(3, \frac{4}{10}), \quad \bar{Y} \sim N(-3, \frac{6}{15}). \] Since the samples are independent: \[ \bar{X} + \bar{Y} \sim N(0, \frac{2}{5}). \] 2. Denominator \( S \): The sample standard deviation \( S \) follows a scaled \( \chi^2 \) distribution with 9 degrees of freedom. 3. Distribution of \( U \): By definition, the statistic \( U \) follows a \( t \)-distribution with 9 degrees of freedom.