The mean and variance of the times $$ x_1, x_2, ..., x_N$$ used in performing N similar tasks, are: 14 and 2.89. The cost to perform each task is $$y_i = 20 + 0.5x_i + 0.1x_i^2 $$ Find the mean of the costs
I am clear about this, $$a)\bar{x}=\frac{\sum_{i=1}^N{x_i}}{N}=14$$ $$b)\sigma ^2=\frac{\sum_{i=1}^N{(x_i- \bar{x}) ^2}}{N}=2.89$$ but I do not know how to enter $$y_i$$ there
edited what is in bold
Write the formula for $\overline y$ the same way you have for $\overline x$. Plug in the expression you have for $y_i$ to get the sum in terms of $x_i$. After some manipulation you should be able to write it in terms of $\overline x$ and $\overline{x^2}$ and finally in terms of $\overline x$ and $\sigma^2$
$\sigma_x ^2=\frac{\sum_{i=1}^N{(x_i- \bar{x}) ^2}}{N}= \frac{\sum_{i=1}^N{(x_i^2- 2\bar{x}x_i+\bar x^2) }}{N}= \frac{\sum_{i=1}^N{(x_i^2- \bar x^2) }}{N}=\bar {x^2}-(\bar x)^2$
You know $\sigma_x^2$ and $(\bar x)^2$, so now you know $\bar{x^2}$ to use in the computation of $\bar y$