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

a. We wish to test the hypothesis that there is no relationship between the number of medals won and \(G D P\) against the alternative there is a positive relationship. State the null and alternative hypotheses in terms of the model parameters.

b. What is the test statistic for part (a) and what is its distribution if the null hypothesis is true?

c. What happens to the distribution of the test statistic for part (a) if the alternative hypothesis is true? Is the distribution shifted to the left or right, relative to the usual \(t\)-distribution?

d. For a test at the \(1 \%\) level of significance, for what values of the \(t\)-statistic will we reject the null hypothesis in part (a)? For what values will we fail to reject the null hypothesis?

e. Carry out the \(t\)-test for the null hypothesis in part (a) at the \(1 \%\) level of significance. What is your economic conclusion? What does \(1 \%\) level of significance mean in this example?

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MEDALS = b + bGDPB = 7.61733 +0.01309GDPB (se) (2.38994) (0.00215) (XR3.1)