Value of Linear Combination of Missing Observations

Given the dataset of 10 observations $x_1, x_2, \dots, x_{10}$ with mean $\mu = 5$ and variance $\sigma^2 = 7$:

$\sum_{i=1}^{10} x_i = 10 \times 5 = 50$

$\sum_{i=1}^{10} x_i^2 = 10 \times (7 + 5^2) = 320$

For the first 8 observations $x_1, x_2, \dots, x_8$ with mean $\mu_8 = 4$ and variance $\sigma_8^2 = 3.5$:

$\sum_{i=1}^{8} x_i = 8 \times 4 = 32$

$\sum_{i=1}^{8} x_i^2 = 8 \times (3.5 + 4^2) = 156$

Subtracting the sums for the first 8 observations from the total sums gives equations for $x_9$ and $x_{10}$:

$x_9 + x_{10} = 50 - 32 = 18$

$x_9^2 + x_{10}^2 = 320 - 156 = 164$

Using the identity $(x_9 + x_{10})^2 = x_9^2 + x_{10}^2 + 2x_9x_{10}$:

$18^2 = 164 + 2x_9x_{10} \implies x_9x_{10} = 80$

Solving the system $x_9 + x_{10} = 18$ and $x_9x_{10} = 80$ with the condition $x_9 < x_{10}$ yields:

$x_9 = 8, \quad x_{10} = 10$

Thus, the value of $3x_9 + 2x_{10}$ is:

$3(8) + 2(10) = 44$