1 Inference for Linear Regression

Is this assignment we will develop the theory for making inferences about our regression model:

\[y = \beta_0 + \beta_1x+\epsilon\]

Where \(\beta_0\) and \(\beta_1\) are unknown parameters (corresponding to y-intercept and slope of linear model), and \(\epsilon\) is the random error.

To test whether x makes any contribution to the prediction of y, we will use the hypothesis:

\(H_o: \beta_1=0\)
\(H_a: \beta_1\neq0\)

So what is the sampling distribution of our estimate \(\hat\beta_1\)??

If we make our four assumptions about the random error \(\epsilon\), we can prove: \[\hat\beta_1 \sim N \Big(\beta_1, \frac{\sigma}{\sqrt{SS_{xx}}} \Big)\]

Where \(SS_{xx} = \sum_i^n (x_i-\bar{x})^2 = (n-1)s^2_x\).

  1. Show expected value of our slope estimate, \(\hat{\beta_1}\), is \(\beta_1\):

\[\mu_\hat{\beta_1}=E(\hat\beta_1)=\beta_1\]

  1. Show variance of Show the expected value of our slope estimate, \(\hat{\beta_1}\), is or:

\[\sigma^2_\hat{\beta_1}=Var(\hat\beta_1)=\frac{\sigma^2}{SS_{xx}}\]

2 Extra Credit

  1. Show that \(E(\hat{\beta_0})=\beta_0\).

  2. Show that \(Var(\hat{\beta_0})=\frac{\sigma^2}{n}\frac{\sum_{i=1}^n x_i^2}{SS_{xx}}\).