Question

a scatterplot shows the relationship between the number of absences and the gpa (on a 4 - point scale) for a large number of high school students.
if the point labeled a is removed, which of the following statements would be true?
(a) the slope of the least - squares regression line would decrease and the strength of the association would increase.
b) the slope of the least - squares regression line would decrease and the strength of the association would decrease.
c) the slope of the least - squares regression line would increase and the strength of the association would remain the same.
(d) the slope of the least - squares regression line would increase and the strength of the association would increase.
e) the slope of the least - squares regression line would increase and the strength of the association would decrease.

Explanation:

Step1: Analyze the effect of the out - point on the slope

The point labeled \(A\) is an outlier. Since the relationship between the number of absences and GPA is negative (as the number of absences increases, GPA decreases). The outlier \(A\) has a low number of absences but a relatively high GPA. When we remove this outlier, the data points will be more concentrated around a steeper negative - trend line. Mathematically, if we consider the formula for the slope of the least - squares regression line \(b = r\frac{s_y}{s_x}\) (where \(r\) is the correlation coefficient, \(s_y\) is the standard deviation of the \(y\) - variable (GPA), and \(s_x\) is the standard deviation of the \(x\) - variable (number of absences)). Removing the outlier makes the data more linearly related (in a negative direction), which increases the magnitude of the negative correlation coefficient \(r\) (since \(r\) is negative for a negative relationship). So the slope \(b\) (which is negative) will become more negative (its magnitude increases).

Step2: Analyze the effect of the out - point on the strength of the association

The strength of the association is measured by the absolute value of the correlation coefficient \(|r|\). The outlier \(A\) weakens the linear association. When we remove the outlier, the data points follow the negative linear trend more closely. So the absolute value of the correlation coefficient \(|r|\) increases, which means the strength of the association increases.

Answer:

D. The slope of the least - squares regression line would increase and the strength of the association would increase.