The data set faithful in $R$ package faraway contains the variables eruptions which is the average number of eruptions by the Old Faithful Geyser at any given point in time $($in mins$),$ and waiting which is the waiting time before the next eruptions $($in mins$)($H$$rdle$,$ W$.(1991).{?}^1.$ The descriptions of these variables are presented in Table $1.$

Table $1$: Variables and their description in the faithful database

$\backslash$table$[[$Variable$,$Description$],[$eruptions$,$Eruption time in mins$],[$waiting$,$Waiting time to next eruption $($in mins$)]]$

$($a$)$ Obtain a kernel density estimate for Eruption time in mins, eruptions, and plot it$.$ You are free to decide on whatever kernel and bandwidth you like, but specify which ones you used.

$($b$)$ Obtain a parametric density estimate assuming that Eruption time in mins, eruptions, follow a normal distribution. Overlay this density estimate with your estimate from $($a$).$ Comment on the two density estimates.

$($c$)$ Repeat the questions in part $($a$)$ and $($b$)$ for the variable Waiting time to next eruption $($in mins$),$ waiting. Comment on the two density estimates.

$($d$)$ For the variable Waiting, time to next eruption $($in mins$),$ waiting, compare the density estimates produced by smoothing parameters which are chosen by the normal optimal, Sheather$-$Jones and cross$-$validatory approaches. In each case, try to make your own judgement on a good value of smoothing parameter and compare this with the one chosen by the automatic methods. Where there are differences, try or identify the cause.

$($e$)$ Apply the test of normality to the variables Waiting time to next eruption $($in mins$),$ waiting, and Eruption time in mins, eruptions, and construct an associated reference band as a simple means of investigating whether the asymmetry in the density estimate for each variable is more than would be expected from normal random variation.

$($f$)$ In part $($e$),$ explore the effect of changing the smoothing parameter on the relationship between the reference bands and the density estimate.