Inferences about regression

Y. The X values can be arbitrarily chosen for a simulation, and then can remain the same for each data set created in the simulation. The linear regression model describes the distribution of Y values for particular values of X. So, X divides the units into groups, much like the grouping variable in a t-test.

X values that are farther apart from one another

All three. When the residual variance is big, the points are further from the line, and so we are less sure of our estimate of the slope (and the variance of β₁-hat increases). When the X values are spread far apart, we have more information about the slope of the line than we would if all the points had similar values of X (and the denominator of the variance of β₁-hat increases, so that the variance of β₁-hat decreases). As the sample size increases, we have more information about the slope (and the summation in the denominator of the variance of β₁-hat increases, so that the variance of β₁-hat decreases). If the sample size was equal to infinity, the variance of β₁-hat would be 0, because we would have information about all the units in the population, and our estimate of β₁-hat would be exactly equal to β₁.

The slope. We usually don't particularly care what the mean of Y is when X is 0 (this is the intercept). We are much more interested in how the mean of Y changes when X increases by specified amount (this is the slope).