Question

Math

Posted 3 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 3 months ago

Solution by Steps

step 1

Identify the outlier in the scatterplot. The point $(66,15)$ is an outlier because it is significantly distant from the other points and the regression line

step 2

Understand the effect of an outlier on the least-squares regression line. An outlier can disproportionately affect the slope and y-intercept of the regression line, potentially skewing the analysis

step 3

Predict the changes after removing the outlier. Removing the outlier $(66,15)$ will likely result in a regression line that better fits the remaining data points

step 4

Assess the potential effects on the coefficient of determination $r^{2}$. Since the outlier is above the regression line, removing it would likely increase the $r^{2}$ value, indicating a better fit for the data

step 5

Evaluate the potential effects on the standard deviation of the residuals. Without the outlier, the residuals (differences between observed and predicted values) are likely to be smaller, which would decrease the standard deviation of the residuals

step 6

Consider the potential effects on the correlation coefficient $(r)$. If the outlier is causing the data to appear less correlated, removing it could result in a correlation coefficient $(r)$ that is closer to 1 or -1, depending on the direction of the correlation

[question number] Answer

A: Incorrect, B: Correct, C: Incorrect

Key Concept

Effect of Removing an Outlier on Regression Analysis

Explanation

Removing an outlier from a dataset can increase the coefficient of determination, decrease the standard deviation of the residuals, and potentially bring the correlation coefficient closer to 1 or -1, reflecting a stronger linear relationship.

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