1. ## Regression analysis

In a regression analysis using the usual model for Normal data, ten y values are observed, one at each of x=1,2,3,...10. Compute the estimated correlation between $\displaystyle \alpha$ and $\displaystyle \beta$.

(alpha and beta have hats on but I couldn't get the latex to work for that)

You can work out $\displaystyle S_{xx}$ from the given information. But I'm not sure where to begin to compute the correlation.

2. $\displaystyle r^2={ SS_{xy}\over SS_{xx} SS_{yy}}$

and if you're not used to that notation...

$\displaystyle { \sum_j\sum_i(x_i-\bar x)(y_j-\bar y) \over \sum_i(x_i-\bar x)^2\sum_j(y_j-\bar y)^2}$

which is the sample covariance squared over the sample variances (using n instead of n-1, the MLEs)
where we cancel the n's via algebra.

3. using $\displaystyle r^2= \frac{Cov(\alpha,\beta)^2}{Var(\alpha)Var(\beta)}$ and the definitions of those in terms of x and $\displaystyle S_{xx}$ i got...

$\displaystyle r^2=\frac{n\bar{x}^2}{S_{xx}}+1$ EDIT: whoops that logic is wrong, figured it out properly now.

would this be correct? Then I can just use the values calculated of xbar and Sxx?

4. In the case of simple linear regression, $\displaystyle r = \sqrt{R^{2}} = \sqrt{\frac{SSR}{SSTO}} = \sqrt{1-\frac{SSE}{SSTO}}$