A researcher uses the regression equation $\hat{y}=5+0.6x$ to predict the number of hours studied ($y$) based on the number of classes attended ($x$). The standard error of the estimate is $s_e=2.5$, the mean number of classes attended is $\bar{x}=15$, $n=12$, and $\sum (x_i-\bar{x}{)}^2=120$. Construct a $99\%$ prediction interval for a student who attended $20$ classes.