A research team collected data on $n=450$ students in a statistics course. The observed average final grades was $520,$ with an observed standard deviation of $127\mathrm{.}0($the divisor in the estimated variance was $n-1).$ The average practice final grades was $390,$ with an observed standard deviation of $96\mathrm{.}0.$ The correlation coefficient between the practice final grades and the final grades was $0\mathrm{.}53.$

$($a$)$ Report the analysis of variance table $($ANOVA$)$ and result of the test of the null hypothesis that the slope and the intersect of the regression line of final grades on practice final grades are both zero against the alternative that at least one of them is not zero. Use the $0\mathrm{.}10,0\mathrm{.}05,$ and $0\mathrm{.}01$ levels of significance.

$($b$)$ Use the least$-$squares prediction equation to predict the final examination score of a student who scored $550$ on the first examination. Give the $99\%$ prediction interval for the final examination score of this student.