Consider a data consisting of 10 observations $x_1$, $x_2$, ..., $x_{10}$, whose mean is 5 and variance is 7. If the mean and the variance of the first 8 observations $x_1$, $x_2$, ..., $x_8$ are 4 and 3.5, respectively, and $x_9 < x_{10}$, then the value of $3x_9 + 2x_{10}$ is _________.

$\text{Whole data (10): mean } \bar{x}=5,\ \text{variance}=7$ $\text{First 8: mean}=4,\ \text{variance}=3.5$ $x_9 < x_{10}$ $\text{To find: } 3x_9 + 2x_{10}$ $1.\ \sum x,\ \sum x^2 \text{ for all } 10$ $2.\ \text{last two}: x_9+x_{10},\ x_9^2+x_{10}^2$ $3.\ \text{find } x_9,\ x_{10} \Rightarrow 3x_9+2x_{10}$ $\text{variance} = \frac{\sum x^2}{n} - (\bar{x})^2$ $\sum x = 10 \times 5 = 50$ $\sum x^2 = 10(7 + 25) = 320$ $\sum_{i=1}^{8} x_i = 8 \times 4 = 32$ $\sum_{i=1}^{8} x_i^2 = 8(3.5 + 16) = 156$ $x_9 + x_{10} = 50 - 32 = 18$ $x_9^2 + x_{10}^2 = 320 - 156 = 164$ $(x_9 + x_{10})^2 = 18^2 = 324$ $= (x_9^2 + x_{10}^2) + 2x_9 x_{10} = 164 + 2x_9x_{10}$ $\Rightarrow x_9 x_{10} = 80$ $t^2 - 18t + 80 = 0$ $t = \frac{18 \pm 2}{2} = 10,\ 8$ $x_9 = 8,\quad x_{10} = 10$ $3x_9 + 2x_{10} = 3(8) + 2(10) = 24 + 20$ $= 44$