There were 64 countries who competed in the 1992 Olympics and won at least one medal. For

Question:

There were 64 countries who competed in the 1992 Olympics and won at least one medal. For each of these countries, let MEDALS be the total number of medals won, $P O P M$ be population in millions, and GDPB be GDP in billions of 1995 dollars. Excluding the United Kingdom, and using $N=63$ observations, the model MEDALS $=\beta_{1}+\beta_{2} \ln (P O P M)+\beta_{3} \ln (G D P B)+e$ was estimated as

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a. Given assumptions MR1-MR6 hold, interpret the coefficient estimates for $\beta_{2}$ and $\beta_{3}$.

b. Interpret $R^{2}$.

c. 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 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. [The standard error for $b_{1}+\ln (58) \times b_{2}+\ln (1010) \times b_{3}$ is 4.22196.]

f. The United Kingdom won 20 medals in 1992. Is the model a good one for predicting the mean number of medals for the United Kingdom? What is an approximate $p$-value for a test of $H_{0}: \beta_{1}+\ln (58) \times \beta_{2}+\ln (1010) \times \beta_{3}=20$ versus $H_{1}: \beta_{1}+\ln (58) \times \beta_{2}+\ln (1010) \times \beta_{3} eq 20$ ?
g. Without doing any of the calculations, write down the expression that is used to compute the standard error given in part (e).

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