Answer: $$ \begin{bmatrix} 432.86 \\ 364.29 \end{bmatrix} \sim \text{N} \left( \begin{bmatrix} \mu_x \\ \mu_y \end{bmatrix}, \begin{bmatrix} 4058.71 & 1106.37 \\ 1106.37 & 914.81 \end{bmatrix} \right) $$

## Explanation To determine the correct representation of the sample means distribution, we need to calculate the sample means and the covariance matrix. ### Step 1: Calculate Sample Means **For X:** Sum of X = 700 + 130 + 140 + 200 + 230 + 250 + 270 + 340 + 400 + 620 + 620 + 760 + 650 + 750 = 6,060 Sample size n = 14 Mean of X = 6,060 / 14 = 432.86 ✓ **For Y:** Sum of Y = 318 + 304 + 317 + 305 + 309 + 307 + 316 + 309 + 315 + 327 + 450 + 324 + 500 + 699 = 5,100 Mean of Y = 5,100 / 14 = 364.29 ✓ ### Step 2: Calculate Sample Variances and Covariance **Variance of X:** Using the formula: Var(X) = Σ(x_i - μ_x)² / (n-1) After calculations: Var(X) ≈ 56,821.94 Sample variance for distribution of means = Var(X)/n = 56,821.94 / 14 ≈ 4,058.71 ✓ **Variance of Y:** Var(Y) = Σ(y_i - μ_y)² / (n-1) ≈ 12,807.34 Sample variance for distribution of means = Var(Y)/n = 12,807.34 / 14 ≈ 914.81 ✓ **Covariance:** Cov(X,Y) = Σ[(x_i - μ_x)(y_i - μ_y)] / (n-1) ≈ 15,489.18 Sample covariance for distribution of means = Cov(X,Y)/n = 15,489.18 / 14 ≈ 1,106.37 ✓ ### Step 3: Verify the Distribution The correct representation is: $$ \begin{bmatrix} 432.86 \\ 364.29 \end{bmatrix} \sim \text{N} \left( \begin{bmatrix} \mu_x \\ \mu_y \end{bmatrix}, \begin{bmatrix} 4058.71 & 1106.37 \\ 1106.37 & 914.81 \end{bmatrix} \right) $$ This matches **Option C** exactly. The covariance matrix is symmetric, so the off-diagonal elements should be the same (1,106.37), which is correctly shown in Option C. **Why other options are incorrect:** - Option A: Wrong Y mean (264.29) and wrong covariance matrix symmetry - Option B: Wrong Y mean (464.29) and wrong covariance matrix symmetry - Option D: Wrong Y mean (324.29) and wrong variance/covariance values

Author: Tanishq Prabhu