Unusual points Each of the four scatterplots that follow shows a cluster of points and one “stray” point. For each, answer these questions: 1) In what

emancipezN 2021-03-02 Answered

Unusual points Each of the four scatterplots that follow shows a cluster of points and one “stray” point. For each, answer these questions:
1) In what way is the point unusual? Does it have high leverage, a large residual, or both?
2) Do you think that point is an influential point?
3) If that point were removed, would the correlation be- come stronger or weaker? Explain.
4) If that point were removed, would the slope of the re- gression line increase or decrease? Explain
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Expert Answer

hajavaF
Answered 2021-03-03 Author has 90 answers

a.
1.
Residual:
The residual corresponding to a predictor variable is given as the difference between actual value of the response variable and the predicted value. That is, e=y−yˆ, where y be the actual value of the response variable and yˆ be the predicted value of the response variable for same predictor variable.
Leverage:
An observation, whose predictor variables values (x values) are far from the mean of the predictor variables values (x values) is called as leverage point. Leverage point pulls the regression line to it and has a large effect on the regression line. An observation having high leverage has small residual.
The point pulls the regression line to it and has a large effect on the regression line. In addition, the difference between observed and predicted value of response variable corresponding to this point is high.
Thus, the point has a high leverage with a high residual.
2.
Influential point:
A point, which does not belong in a data set and the omission of which from the data results in a very different regression model, is called as influential point.
The point is far from the mean of the explanatory variable. Moreover, the omission of the point from the data results in a very different regression model as it reinforces the association. In addition, including the point scatterplot shows an overall positive direction that is not the actual direction.
Thus, the point is an influential point.
3.
Association:
Association between two variables implies that if two variables are associated or related then the value of one variable gives information about the value of the other variable.
Correlation measure the linear relationship between two variables.
The point supports the positive association. Removing of this point it would weaken the association.
As a result of this the correlation would become weaker. Thus, removing the point result in a weaker correlation.
4.
In a linear regression model y^=b0+b1x, where yˆ be the predicted values of response variable and x be the predictor variable, the b1b1 be the slope and b0b0 be the intercept of the line.
Slope gives the rapidly change of y with respect to x and slope estimate is given as,
b1=r((sy)/(sx)), where r be the correlation between x and y, sy be the standard deviation of y and sx be the standard deviation of x.
The slope of the regression line would increase from negative slope to a slope near 0.
Thus, if the point were removed, would the slope of the regression line would be nearly flat.
b.
1.
The point pulls the regression line to it and has a large effect on the regression line.
Thus, the point has a high leverage with a small residual.
2.
The point is far from the actually scattered points and direction of scatterplot is positive due to this point when the points are actually scattered. Moreover, the omission of the point from the data results in a very different regression model.
Thus, the point is an influential point.
3.
Removing of this point it would weaken the association. Except the point there would be little evidence of linear association.
As a result of this the correlation would become weaker.
Thus, removing the point result in a weaker correlation.
4.
As the point is not influential, thus removing of the point is not result in a very different regression result.
The slope of the regression line would increase from negative slope to a slope near 0.
Thus, if the point were removed, would the slope of the regression line would be nearly flat.
c.
1.
The point does not pull the regression line to it and has not a large effect on the regression line. The difference between observed and the predicted value of response variable corresponding to that point is quite high.
Thus, the point has a little leverage with a high residual.
2.
The point is close to the mean of the explanatory variable. Moreover, the omission of the point from the data results in not a different regression model.
Thus, the point is not influential point.
3.
Removing of this point it would reinforce the association as the point detracts from the overall pattern.
Thus, removing the point result in a slightly stronger correlation, decreasing to become negative.
4.
As the point is not influential, thus removing of the point is not result in a very different regression result.
The slope of the regression line would not be affected.
Thus, if the point were removed, would the slope of the regression line would be remain same.
d.
1.
The point pulls the regression line to it and has a large effect on the regression line.
Thus, the point has a high leverage with a small residual.
2.
The point is far from the mean of the explanatory variable gives the high leverage. Moreover, the omission of the point from the data results in not a very different regression model as it reinforces the association.
Thus, the point is not an influential point.
3.
The point supports the negative association. Removing of this point it would weaken the association.
As a result of this the correlation would become weaker.
Thus, removing the point result in a weaker correlation.
4.
As the point is not influential, thus removing of the point is not result in a very different regression result.
Thus, if the point were removed, would the slope of the regression line remain same

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