Question

Math

Posted 5 months ago

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The scatterplot below displays a set of bivariate data along with its leastsquares regression line.
Consider removing the point $(66,15)$ and calculating a new least-squares regression line.
What effect(s) would removing this point have?
Choose all answers that apply:
A The coefficient of determination $\left(r^{2}\right)$ would decrease.
B The standard deviation of the residuals would decrease.
C The correlation coefficient $(r)$ would get closer to -1 .
```

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Answer from Sia

Posted 5 months ago

Solution by Steps

step 1

Identify the outlier in the scatterplot

step 2

Recognize that the point (66,15) lies above the regression line and is an outlier

step 3

Understand that removing an outlier can affect the regression line, the correlation coefficient, and other statistical measures

step 4

Predict that removing the outlier will likely result in a regression line that better fits the remaining data points

step 5

Anticipate that the correlation coefficient $r$ will likely increase in absolute value, getting closer to -1 if the outlier is in the direction of the positive correlation

step 6

Expect that the standard deviation of the residuals will decrease because the outlier that is contributing to a larger spread of residuals is removed

step 7

Consider that the coefficient of determination $r^2$ will likely increase, reflecting a better fit of the regression line to the data points without the outlier

Answer

B The standard deviation of the residuals would decrease.
C The correlation coefficient $(r)$ would get closer to -1.

Key Concept

Impact of Removing an Outlier on Regression Analysis

Explanation

Removing an outlier from a set of data points can lead to a decrease in the standard deviation of the residuals and an increase in the absolute value of the correlation coefficient, reflecting a stronger linear relationship between the variables.

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