Standard error of slope

When b1 is used as an estimate of β1, the estimation error has a normal distribution,

error in estimate of β1   =   (b1 − β1)   ~   normal ( 0,  σb1 )

This standard deviation is the standard error of the estimate,

where sx is the standard deviation of X. Since σ is unknown, we must replace it with an estimate from the data to obtain a numerical value for the standard error,

Example

The estimated error distribution gives in indication of how close our least squares estimate, b1 = 0.468, is likely to be to the population regression slope, β1.