Repeat Problem 2.23 using only 10 observations in each sample, drawing one observation from each level $x=1,2,3, \ldots, 10$. What impact does using $n=10$ have on the questions asked in Problem 2.23? Compare the lengths of the CIs and the appearance of the histograms.

Data From Problem 2.23

Consider the simple linear regression model y=50+10x+ε$\mathrm{????}=50+10\mathrm{????}+\mathrm{????}$ where ε$\mathrm{????}$ is NID (0,16)$(0,16)$. Suppose that n=20$\mathrm{????}=20$ 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,…,10$\mathrm{????}=1,1.5,2,\dots ,10$ for each sample.

a. For each sample compute the least-squares estimates of the slope and intercept. Construct histograms of the sample values of ^β0${\mathrm{????}^}_0$ and ^β1${\mathrm{????}^}_1$. Discuss the shape of these histograms.

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

c. For each sample, compute a 95$95$ CI on the slope. How many of these intervals contain the true value β1=10${\mathrm{????}}_1=10$ ? Is this what you would expect?

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