The least-squares regression line is the line that best fits a bivariate dataset in the sense of minimizing the sum of the squares of the vertical distances from each point in the dataset to the line. Denote the points in the dataset (
. Assume that
follows a normal distribution whose mean is a linear function of
(with unknown slope and intercept) and whose standard deviation is a constant function of
. Then a confidence interval for the expected value of
can be constructed using standard techniques. As the expected value of
is a function of
, the endpoints of this interval will be as well. When plotted as a function of
, these endpoints form "confidence bands" between which runs the regression line.
A confidence interval for the
value associated with a new
value (as opposed to a confidence interval for the mean of all such
values) is called a prediction interval. Its endpoints are also functions of
, which when plotted form "prediction bands". As individual
values vary more than their mean, the prediction bands are wider than the confidence bands.
Browse all topics
