Statistics

We know the mean of 10 observations is 5. So, the total sum of the 10 observations is:

$\frac{x_1+x_2+x_3+\ldots+x_8+x_9+x_{10}}{10}=5$

Multiplying both sides by 10 gives:

$x_1+x_2+x_3+\ldots+x_8+x_9+x_{10}=50$

The mean of the first 8 observations is 4, so their total sum is:

$\frac{x_1+x_2+\ldots+x_8}{8}=4$

Therefore,

$x_1+x_2+\ldots+x_8=32$

Subtracting, we get:

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

Next, we use the formula for variance of the 10 observations:

$\text{Variance} = \frac{\sum\limits_{i=1}^{10} x_i^2}{10} - (\text{mean})^2$

Substitute the given values to find the total of squares:

$\frac{\sum\limits_{i=1}^{10} x_i^2}{10} - 25 = 7$

$\sum\limits_{i=1}^{10} x_i^2 = 320$

Similarly, for the first 8 observations:

$\frac{\sum\limits_{i=1}^{8} x_i^2}{8} - 16 = 3.5$

$\sum\limits_{i=1}^{8} x_i^2 = 156$

Now, subtract the two results:

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

We already know:

$x_9 + x_{10} = 18$

Expanding $ (x_9 + x_{10})^2 $, we get:

$x_9^2 + x_{10}^2 + 2x_9x_{10} = 324$

Substitute $x_9^2 + x_{10}^2 = 164$:

$164 + 2x_9x_{10} = 324$

$x_9x_{10} = 80$

Now, we have two equations:

$x_9 + x_{10} = 18$

$x_9x_{10} = 80$

Solving the quadratic $t^2 - 18t + 80 = 0$, we get $t = 8$ and $10$. Given $x_9 < x_{10}$, so $x_9 = 8$ and $x_{10} = 10$.

Finally,

$3x_9 + 2x_{10} = 3(8) + 2(10) = 44$