Problem $1$ Consider the problem of studying the students' test score on a certain course based on three

predictors: hours of studying $(x_1),$ number of practice tests taken $(x_2),$ and average grade in other classes

$(x_3).$ The data is shown in Fig. $1.$

Figure $1$: PCA

$($a$)$ Compute the multivariate mean

$($b$)$ Compute the mean$-$centered data matrix

$($c$)$ Find the covariance matrix of the centered data

$($d$)$ Compute the eigenvalues of the covariance matrix

$($e$)$ Compute the principal components $($please$,$ explicitly show correspondence between eigenvalues and

principal components$)$ and transformation matrix

$($f$)$ Compute PCA transformation of the original data matrix

$($g$)$ How much variance is explained by new "hours of studying"?