Question

1. a scatterplot and a least - squares regression line are shown in the figure below. if the point (20, (the circled point) is removed from the data set, which of the statements below is true?

(a) the slope will decrease and the y - intercept will increase.
(b) the slope will decrease and the y - intercept will decrease.
(c) the slope will increase and the y - intercept will increase.
(d) the slope will increase and the y - intercept will decrease.
(e) no conclusion can be drawn since the coordinates of the other points are not known.

Explanation:

Step1: Analyze the effect of the circled point on the slope

The circled point has a large \(x\) - value (\(x = 20\)) and a \(y\) - value that is higher than what the regression line (based on the other points) would predict. When calculating the slope \(b=\frac{\sum_{i = 1}^{n}(x_{i}-\bar{x})(y_{i}-\bar{y})}{\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}}\), the numerator \(\sum_{i = 1}^{n}(x_{i}-\bar{x})(y_{i}-\bar{y})\) and the denominator \(\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}\) are affected. The point \((20,y)\) (where \(y\) is relatively large) makes the slope smaller. Removing it will make the slope increase.

Step2: Analyze the effect of the circled point on the \(y\) - intercept

The \(y\) - intercept \(a=\bar{y}-b\bar{x}\). Since the slope \(b\) will increase (from Step 1) and \(\bar{x}\) and \(\bar{y}\) are recalculated without the out - lying point. Intuitively, the original regression line was “pulled down” at the higher \(x\) values. When we remove the point \((20,y)\) (which was an over - prediction in terms of the line's fit to the other points), the new regression line will have a smaller \(y\) - intercept.

Answer:

C. The slope will increase and the y - intercept will decrease.