Question 12

For 10 observations: $\sum x_i = 10 \times 5 = 50$. Variance: $\frac{\sum x_i^2}{10} - 5^2 = 7 \Rightarrow \sum_{i=1}^{10} x_i^2 = 320$. For first 8 observations: $\sum_{i=1}^8 x_i = 8 \times 4 = 32$. Variance: $\frac{\sum_{i=1}^8 x_i^2}{8} - 4^2 = 3.5 \Rightarrow \sum_{i=1}^8 x_i^2 = 8(19.5) = 156$. Then $x_9 + x_{10} = 50 - 32 = 18$ and $x_9^2 + x_{10}^2 = 320 - 156 = 164$. $(x_9 + x_{10})^2 = x_9^2 + x_{10}^2 + 2x_9x_{10} \Rightarrow 18^2 = 164 + 2x_9x_{10} \Rightarrow 324 - 164 = 2x_9x_{10} \Rightarrow x_9x_{10} = 80$. Solving $t^2 - 18t + 80 = 0$ gives $t = 8, 10$. Since $x_9 < x_{10}$, $x_9 = 8$ and $x_{10} = 10$. $3x_9 + 2x_{10} = 3(8) + 2(10) = 24 + 20 = 44$. Final Answer: 44.