$(8$ marks$)$ In this question, you will fit kNN regression models to the Auto data set to predict $Y=mpg$ using $x=$ horsepower. This data has been divided into training and testing sets: AutoTrain.csv and AutoTest.csv $($download these sets from Learn$).$ The $)R$ function on Learn should be used to answer this question $($you need to run the $kNN$ code before calling the function$).$

$($a$)$ Perform $kNN$ regression with $k=2,5,10,20,30,50$ and $100,($learning from the training data$)$ and compute the training and testing MSE for each value of $k.$

$($b$)$ Which value of $k$ performed best? Explain.

$($c$)$ Plot the training data, testing data and the best kNN model in the same figure. $($The points $()$ function is useful to plot the kNN model because it is discontinuous.$)$

$($d$)$ Describe the bias$-$variance trade$-$off for kNN regression.