91Ó°ÊÓ

Economists have noticed a relationship between an individual's income and consumption pattern. Describe the functional form such a relationship might take and how linear regression might be used to estimate the consumption function.

From a sample of 103 cases, \(r=.80\). (a) Establish the 95 percent confidence interval for the correlation coefficient. (b) Test the hypothesis that \(\rho=.90\).

Show that if \((\mathrm{X}, \mathrm{Y})\) has a bivariate normal distribution, then the marginal distributions of \(\mathrm{X}\) and \(\mathrm{Y}\) are univariate normal distributions; that is, \(\mathrm{X}\) is normally distributed with mean \(\mu_{\mathrm{x}}\) and variance \(\sigma^{2} \mathrm{x}\) and \(\mathrm{Y}\) is normally distributed with mean \(\mu_{\mathrm{y}}\) and variance \(\sigma^{2} \mathrm{y}\).

Use principal component analysis to determine the factor-loadings and the minimum number of common factors that could give rise to the following correlation matrix : \begin{tabular}{|l|l|l|} \hline & \(\mathrm{X}_{1}\) & \(\mathrm{X}_{2}\) \\ \hline \(\mathrm{X}_{1}\) & \(1.00\) & \(.48\) \\ \hline \(\mathrm{X}_{2}\) & \(.48\) & \(1.00\) \\ \hline \end{tabular}