You probably recognized on the previous page that the shape of the sampling distribution for the slope was close to the normal distribution. Indeed, the sampling distribution for the slope is a normal distribution with a mean equal to b and a standard deviation equal to

where

is the standard deviation of the errors from the model of Y using X; that is, the typical distance from an observation to the regression line. Click on the Equations button for the mathematical details.

If we knew the population value of the standard deviation for the sampling distribution of the slope, then we could calculate a z-score and use the normal distribution to identify surprising values. However, we must estimate the sampling distribution's standard deviation. So, just as was the case for the one-group and two-group comparisons, we must instead use the Student t statistic. The standard deviation of the sampling distribution for the slope is easy to estimate from the data. The numerator is based on the typical model squared error--the sum of the error squares divided by the degrees of freedom. This is sometimes called the "mean squared error" (MSE) and is given by:

and the denominator is estimated by the square root of n times the standard deviation for X. Putting everything together,

We will generally use the computer to calculate the Student t for the slope so it is more important to understand the concepts in the above formula than the computational details. The right-most expression tells us when the t for the slope will be large--when the slope b is large, when the number of observations n is large, when the standard deviation of X (the predictor) is large, and when the aggregate error MSE is small (i.e., the data points are close to the best-fitting line).