An experiment was carried out in Tennessee between $1985$ and $1989.$ A group of $5752$ students was followed from kindergarten through third grade. In the experiment children were randomly assigned within schools into two types of classes: small classes with $13-17$ and regular$-$sized classes with $22-25$ students. A description of the available variables is displayed in table $1.$ Using the estimates provided in Table $2,$ answer the following questions:

$($a$)$ Comment on the $R^2$ statistic: would you say that the model fits the data well?

Table $1$: Descriptive statistics

$\backslash$table$[[$Variabile$,$Descrizione,Mean,Median,StDev,Min,Max$],[$math$,\backslash$table$[[$proficiency score in mathematics$],[($in logs$)]],6\mathrm{.}181,6\mathrm{.}182,0\mathrm{.}09714,5\mathrm{.}768,6\mathrm{.}439],[,$dummy, $1$ if child is male,$0\mathrm{.}5135,1\mathrm{.}000,0\mathrm{.}4999,0\mathrm{.}000,1\mathrm{.}000],[$boy$,$dummy, $1$ if child is in a small,$0\mathrm{.}6535,1\mathrm{.}000,0\mathrm{.}4759,0\mathrm{.}000,1\mathrm{.}000],[$small$,$class,,,,,$],[$te$,$teacher years of experience,$9\mathrm{.}306,9\mathrm{.}000,5\mathrm{.}768,0\mathrm{.}000,27\mathrm{.}00],[$freelunch$,$child is entitled to free lunch,$0\mathrm{.}4817,0\mathrm{.}000,0\mathrm{.}4997,0\mathrm{.}000,1\mathrm{.}000],[$tchwhite$,$dummy, $1$ if the teacher is white,$0\mathrm{.}8355,1\mathrm{.}000,0\mathrm{.}3708,0\mathrm{.}000,1\mathrm{.}000],[$black$,$dummy, $1$ if the child is black,$0\mathrm{.}3209,0\mathrm{.}000,0\mathrm{.}4669,0\mathrm{.}000,1\mathrm{.}000],[$both$\_$white,dummy, $1$ if both the teacher and,$0\mathrm{.}6376,1\mathrm{.}000,0\mathrm{.}4807,0\mathrm{.}000,1\mathrm{.}000],[,$the child are white,,,,,$]]$

$($b$)$ Do we have to worry about heteroskedasticity here? Why?

$($c$)$ Comment on the sign and magnitude of the coefficients for the variables boy, small and freelunch:

$($d$)$ It is sometimes claimed that white and Asian children are better at maths than black kids. Do the estimates support this view?