Distribution of the least squares slope and intercept
The least squares estimates b0 and b1 of the two linear model parameters β0 and β1 vary from sample to sample. Each has a distribution that can be described by a probability density function.
The least squares estimates, b0 and b1, have normal distributions that are centered on β0 and β1 respectively.
Sampling variability of the least squares line
The diagram below shows a normal linear model and a data set that is sampled from this model. The least squares line is shown in blue.
Click Take sample a few times to observe the variability in the least squares lines.
Click the checkbox Accumulate then click Take sample about 10 times. The variability of the least squares lines is shown on the right. (Click on any line to show the sample to which it belongs.)
Sampling distributions of b0 and b1
We have not yet seen ways to describe the variability of complex summaries such as least squares lines. It is much easier to describe the separate distributions of b0 and b1.
Click the checkbox Accumulate then click Take sample several times. The variability in each parameter estimate is shown in a stacked dot plot. (Click on any cross on the right to show the sample to which it belongs.)
Each parameter estimate has a univariate distribution. Click the checkbox above to superimpose its theoretical normal distribution on each stacked dot plot.