Suppose you have grouped data and the $i$ th response $y_i$ is the average of $n_i$ values.

$($a$)$ You have fit an ordinary least squares regression model ignoring the group sizes $n_i.$ Sketch

the residuals versus $n_i$ plot.

$($b$)$ Write down a regression model which correctly takes the group sizes $n_i$ into account.

$($c$)$ Write down the criterion minimized to fit the above model.

$($d$)$ In the case of a single predictor, the ordinary least squares estimator of the slope $$ in the

no$-$intercept regression model $E(y)=x$ is given by

hat$()=\frac{{\displaystyle \underset{i}{\overset{?}{}}}{x}_i{y}_i}{{\displaystyle \underset{i}{\overset{?}{}}}{x}_i^2}$

The data is

Calculate the estimate of $$ given by $($i$)$ a model fit which ignores the group sizes $n$ and $($ii$)$

a model fit which correctly uses group sizes $n.$