Inference for slopes
A sample’s fitted line rises. What does its slope say about the true rate of change across the whole population?
Where you are 4 inside
What this area is about
A least-squares line \hat{y} = a + bx describes one sample. Behind it sits a population line \mu_y = \alpha + \beta x, and another sample would give another slope b. This area does for the slope \beta what inference for means did for \mu: a range of plausible values, or a verdict on a claimed value, usually \beta = 0, no linear relationship.
How these fit together
The model. The sampling distribution of the slope shows how b varies from sample to sample around \beta, which is what gives it a standard error.
The conditions. The conditions for regression inference are five, spelling LINER: a linear trend; independent observations, with the sample at most 10\% of the population when drawn without replacement; normal responses about the line, judged from residuals with no strong skew and no outliers; equal spread at every x; and a random sample or random assignment.
The procedures. The interval for a slope estimates \beta as b \pm t^* SE_b, read from computer output, with n - 2 degrees of freedom. The test for a slope asks whether a sample slope this far from 0 could arise by chance if \beta = 0.
A path through it
The sampling distribution of the slope first, then the conditions, then the interval, then the test.
Finish here and a fitted slope becomes evidence about how y changes with x across the whole population, not just a description of the sample.