6.9 Regression Model Sampling Distributions

Prior to this point, we’ve looked at the probability theory for sample means. This is because the equations are easier to write down and work with.

But, how does all of this apply to linear models and regression coefficients?

Since the estimates for linear models are very similar to means, the sampling distribution of a sample regression coefficient is approximately Normal with mean = population regression coefficient, \(\beta_j\), and standard deviation, \(SD(\hat{\beta_j})\).

• When the sample size is small \(n<30\), we need to consider William Gosset’s work because \(\frac{\hat{\beta_j} - \beta_j}{SE(\hat{\beta_j})}\) is not quite Normally distributed!