There were 64 countries who competed in the 1992 Olympics and won at least one medal. For each of these countries, let MEDALTOT be the total number of medals won, \(P O P\) be population in millions, and GDP be GDP in billions of 1995 dollars.

a. Use the data file olympics 5, excluding the United Kingdom, and use the \(N=63\) remaining observations. Estimate the model MEDALTOT \(=\beta_{1}+\beta_{2} \ln (P O P)+\beta_{3} \ln (G D P)+e\) by OLS.

b. Calculate the squared least squares residuals \(\hat{e}_{i}^{2}\) from the regression in (a). Regress \(\hat{e}_{i}^{2}\) on \(\ln (P O P)\) and \(\ln (G D P)\). Use the \(F\)-test from this regression to test for heteroskedasticity at the \(5 \%\) level of significance. Use the \(R^{2}\) from this regression to test for heteroskedasticity. What are the \(p\)-values of the two tests?

c. Reestimate the model in (a) but using heteroskedasticity robust standard errors. Using a 10\% significance level, test the hypothesis that there is no relationship between the number of medals won and GDP against the alternative that there is a positive relationship. What happens if you change the significance level to \(5 \%\) ?

d. Using a \(10 \%\) significance level, test the hypothesis that there is no relationship between the number of medals won and population against the alternative that there is a positive relationship. What happens if you change the significance level to 5\%?

e. Use the model in (c) to find point and 95\% interval estimates for the expected number of medals won by the United Kingdom whose population and GDP in 1992 were 58 million and \(\$ 1010\) billion, respectively.

f. The United Kingdom won 20 medals in 1992. Was the model successful in predicting the mean number of medals for the United Kingdom? Using the estimation in (c), with robust standard errors, what is the \(p\)-value for a test of \(H_{0}: \beta_{1}+\ln (58) \times \beta_{2}+\ln (1010) \times \beta_{3}=20\) versus \(H_{1}: \beta_{1}+\ln (58) \times\) \(\beta_{2}+\ln (1010) \times \beta_{3} eq 20\) ?