Statistics

Concept:Use sum and sum-of-squares from given means and variances to find $x_9$ and $x_{10}$, then compute $3x_9+2x_{10}$.Explanation:Mean of 10 observations is 5, so total sum: $\sum_{i=1}^{10} x_i = 10 \times 5 = 50$.Mean of first 8 observations is 4, so their sum: $\sum_{i=1}^{8} x_i = 8 \times 4 = 32$.Subtract: $x_9 + x_{10} = 50 - 32 = 18$.Variance of 10 observations is 7, so $\frac{\sum_{i=1}^{10} x_i^2}{10} - 5^2 = 7$.Thus $\sum_{i=1}^{10} x_i^2 = 10 \times (7 + 25) = 320$.Variance of first 8 is 3.5, so $\frac{\sum_{i=1}^{8} x_i^2}{8} - 4^2 = 3.5$.Thus $\sum_{i=1}^{8} x_i^2 = 8 \times (3.5 + 16) = 156$.Subtract: $x_9^2 + x_{10}^2 = 320 - 156 = 164$.We have $x_9 + x_{10} = 18$ and $x_9^2 + x_{10}^2 = 164$.Square the sum: $(x_9 + x_{10})^2 = 324$ gives $x_9^2 + x_{10}^2 + 2x_9x_{10} = 324$.Substitute $x_9^2 + x_{10}^2 = 164$ to get $164 + 2x_9x_{10} = 324$, so $x_9x_{10} = 80$.Now solve $t^2 - 18t + 80 = 0$: roots are $8$ and $10$.Given $x_9 < x_{10}$, we have $x_9 = 8$, $x_{10} = 10$.Finally, $3x_9 + 2x_{10} = 3(8) + 2(10) = 44$.Answer:$44$