Properties of 95% confidence interval
Confidence intervals for a linear model's slope have the same properties as the confidence intervals that we examined earlier for population means and proportions.
Since the interval is evaluated from random sample data, it will vary from sample to sample. In 95% of such samples, the 95% confidence interval will include the true population slope, but in 5% of samples it will not.
We cannot tell whether or not our single data set is one of the 'lucky' ones.
Simulation
The diagram below shows a sample from a normal linear model in which the true value of β1 is 0.75. (In real data sets, β1 is an unknown value but, by simulating data from a situation where it is known, we can examine the accuracy of our estimates.)