The conditional variance of a bivariate normal is . You can think of the former as a random variable, and the latter as the observed realization of the random variable Let's make a few assumptions about u i . With three or more points, it will normally be impossible to fit the data exactly. Note that . Then plugging in for b , it can be seen that .
Residuals sum to zero.
Recall that if the correlation between two variables is zero, then the covariance between them is also zero. Statistically, we want to know the standard errors of the coefficients. Estimate using . You could then construct confidence intervals for b with the help of a normal distribution table. An hypothesis test is just the flip side of constructing a confidence interval.