Lecture Overview:

• Inferences concerning $\beta_1$ (Ch 2.1)
• Inferences concerning $\beta_0$ (Ch 2.2)
• Interval estimation of $E(Y_h)$ (Ch 2.4)
• Prediction of new observation (Ch 2.5)
• An example with R

Inferences concerning $\beta_1$ (Ch 2.1)

We are interested in the slope parameter $\beta_1$:

e.g.

$$ H_0: \beta_1=0~~~~~~~~~~H_a:\beta_1\neq0 $$

Under SLR settings, when $H_0: \beta_1=0$, then the model becomes: $E(Y)=\beta_0$

Before discussing inferences concerning $\beta_1$, need the sampling distribution of $\beta_1$.

Sampling Distribution of $b_1$

The Sampling Distribution of $b_1$ refers to the indifferent values of b1 that would be obtained with repeated sampling when the levels of the predictor variable X are held constant from sample to sample. (shown in simulation)

The sampling distribution of $b_1$ is normal with mean and variance:

$$ E(b_1)=\beta_1~~~~~~~~~~V(b_1)={\sigma^2\over{S_{XX}}} $$

$b_1$ is a linear combination of $Y_i$