There were 64 countries in 1992 that competed in the Olympics and won at least one medal. Let MEDALS be the total number of medals won, and let GDPB be GDP (billions of 1995 dollars). A linear regression model explaining the number of medals won is \(M E D A L S=\beta_{1}+\beta_{2} G D P B+e\). The estimated relationship is given in equation (XR3.1) in Exercise 3.1.

The estimated covariance between the slope and intercept estimators is -0.00181 and the estimated error variance is \(\hat{\sigma}^{2}=320.336\). The sample mean of \(G D P B\) is \(\overline{G D P B}=390.89\) and the sample variance of \(G D P B\) is \(s_{G D P B}^{2}=1099615\).

a. Estimate the expected number of medals won by a country with \(G D P B=25\).

b. Calculate the standard error of the estimate in (a) using for the variance \ (\widehat{\operatorname{var}}\left(b_{1}\right)+(25)^{2} \widehat{\operatorname{var}}\left(b_{2}\right)+\) (2)(25) \(\widehat{\operatorname{cov}}\left(b_{1}, b_{2}\right)\).

c. Calculate the standard error of the estimate in (a) using for the variance \(\hat{\sigma}^{2}\{(1 / N)+\) \(\left.\left[(25-\overline{G D P B})^{2} /\left((N-1) s_{G D P B}^{2}\right)\right]\right\}\).

d. Construct a \(95 \%\) interval estimate for the expected number of medals won by a country with \(G D P B=25\).

e. Construct a \(95 \%\) interval estimate for the expected number of medals won by a country with \(G D P B=300\). Compare and contrast this interval estimate to that in part (d). Explain the differences you observe.