Confidence interval for the slope
When the least squares slope, b1, is used to estimate β1, the error has a normal distribution,
error in estimate of β1 = (b1 − β1) ~ normal ( 0, σb1 )
This suggests a 95% confidence interval of the form
In practice, we must replace σ in the formula for the standard error with an estimate (based on the sum of squared residuals),
so the constant 1.96 must be replaced by a larger value from the t distribution with (n - 2) degrees of freedom.
A 95% confidence interval for the slope is
Most statistical software will evaluate b1 and its standard error for you when you fit a normal linear model, so it is fairly easy to evaluate the confidence interval in practice — you will not need to use any of the formulae above!
Example
For the example on the previous page, the least squares estimate of the slope and its standard error were:
b1 = 9.27, se (b1) = 1.42
Since there were n = 9 data points, tn − 2 = t7 = 2.365, so a 95% confidence interval for the slope is
We are 95% confident that the expected number of deaths per 100,000 is between 5.9 and 12.6 higher for each unit increase in the exposure index.