Consider the simple linear regression model $y=50+10 x+\varepsilon$ where $\varepsilon$ is NID $(0,16)$. Suppose that $n=20$ pairs of observations are used to fit this model. Generate 500 samples of 20 observations, drawing one observation for each level of $x=1,1.5,2, \ldots, 10$ for each sample.

a. For each sample compute the least-squares estimates of the slope and intercept. Construct histograms of the sample values of $\hat{\beta}_{0}$ and $\hat{\beta}_{1}$. Discuss the shape of these histograms.

b. For each sample, compute an estimate of $E(y \mid x=5)$. Construct a histogram of the estimates you obtained. Discuss the shape of the histogram.

c. For each sample, compute a $95 %$ CI on the slope. How many of these intervals contain the true value $\beta_{1}=10$ ? Is this what you would expect?

d. For each estimate of $E(y \mid x=5)$ in part b, compute the $95 %$ CI. How many of these intervals contain the true value of $E(y \mid x=5)=100$ ? Is this what you would expect?