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What is the effect on the line of regression when one extra point is appended to the original set of data points? This question may arise when one is up-dating time series data, or contemplating gathering more data in any regression-correlational situation. Given a set of data, we will find two vertical lines that have the property that if points from either of these two lines are appended to the original data set and a new line of regression is computed, the new line of regression is changed very slightly. That is, if any point from the vertical line is added to the original data set, the slope of the original line of regression is left unchanged, and if any point from the other vertical line is added to the original data set, the y-intercept of the original line of regression is left unchanged. The original line of regression and the two vertical lines divide the plane into six regions. Any two points, both from the same region, behave qualitatively exactly the same when joined to the original data set. That is, they will both either increase or decrease the slope, and they will both either increase or decrease the y-intercept. Thus, when we append a single point to an original data set, we can determine beforehand what qualitative effect this will have on the line of regression. For example, we could possibly say that the slope will be increased and that the y-intercept will be decreased. In this paper Mathematica is used extensively. Mathematica is used to simplify complicated algebraic expressions, to clarify the theory by giving the reader a look at rather extensive examples, and to graphically display the results of our deliberations. In fact, it was the interplay of conjecture with example which led to many of the results which are given below.
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