Question

Anscombe's Data
Observation	x1	y1		x2	y2		x3	y3		x4	y4
1	10	8.04		10	9.14		10	7.46		8	6.58
2	8	6.95		8	8.14		8	6.77		8	5.76
3	13	7.58		13	8.74		13	12.74		8	7.71
4	9	8.81		9	8.77		9	7.11		8	8.84
5	11	8.33		11	9.26		11	7.81		8	8.47
6	14	9.96		14	8.1		14	8.84		8	7.04
7	6	7.24		6	6.13		6	6.08		8	5.25
8	4	4.26		4	3.1		4	5.39		19	12.5
9	12	10.84		12	9.13		12	8.15		8	5.56
10	7	4.82		7	7.26		7	6.42		8	7.91
11	5	5.68		5	4.74		5	5.73		8	6.89

• Fit a simple linear regression model to each set of (x, y) data, i.e., one model fit to (x₁, y₁), one model fit to (x₂, y₂), one model fit to (x₃, y₃), and one model fit to (x₄, y₄).

• Write down the estimated regression equation for each fitted model, together with the values of the coefficient of determination, r2, and the standard error of the estimate, s=MSE‾‾‾‾‾√.

• For each set of (x, y) data, create a scatterplot of y (vertical) versus x (horizontal) with the estimated regression line added to the plot.

• For each set of (x, y) data, create a scatterplot of the residuals (vertical) versus (horizontal). Based on each plot, do the zero mean and constant variance assumptions about the simple linear regression model error seem reasonable?

• For each set of (x, y) data, create a normal probability plot of the standardized residuals. Based on each plot, does the normality assumption about the simple linear regression model error seem reasonable?

• For each set of (x, y) data, are there any outliers?

• For each set of (x, y) data, are there any high leverage points?

• For each set of (x, y) data, are there any influential points?

  Post a summary of your group’s analysis. What important “big picture” conclusions can you draw from your analysis?

Answer 1

1) for set of (x₁, y₁)

Regression Equation

y1   =   3.00 + 0.500 x1

coefficient of determination R² =66.65%

standard error of the estimate S=1.23660

scatterplot of y1 (vertical) versus x1 (horizontal) with the estimated regression line added to the plot.

a scatterplot of the residuals (vertical) versus (horizontal). Based on each plot

From above graph the zero mean and constant variance assumptions about the simple linear regression model error seem reasonable

a normal probability plot of the standardized residuals

from above graph the normality assumption about the simple linear regression model error seem reasonable

1) for set of (x2, y2)

Regression Equation

y2   =   3.00 + 0.500 x2

coefficient of determination R² =66.62%

standard error of the estimate S=1.2372

scatterplot of y2 (vertical) versus x2 (horizontal) with the estimated regression line added to the plot.

a scatterplot of the residuals (vertical) versus (horizontal). Based on each plot

From above graph the zero mean and constant variance assumptions about the simple linear regression model error seem not reasonable

a normal probability plot of the standardized residuals

from above graph the normality assumption about the simple linear regression model error seem reasonable

3) for (x3,y3)

Regression Equation

y3   =   3.00 + 0.500 x3

coefficient of determination R² =66.6%

standard error of the estimate S=1.2357

scatterplot of y3 (vertical) versus x3 (horizontal) with the estimated regression line added to the plot.

a scatterplot of the residuals (vertical) versus (horizontal). Based on each plot

from above the zero mean and constant variance assumptions about the simple linear regression model error seem not reasonable

a normal probability plot of the standardized residuals

from above graph the normality assumption about the simple linear regression model error seem reasonable

4)

for (x4,y4)

Regression Equation

y4   =   3.00 + 0.500 x4

coefficient of determination R² =66.63%

standard error of the estimate S=1.2361

scatterplot of y3 (vertical) versus x3 (horizontal) with the estimated regression line added to the plot.

a scatterplot of the residuals (vertical) versus (horizontal). Based on each plot

from above the zero mean and constant variance assumptions about the simple linear regression model error seem not reasonable

a normal probability plot of the standardized residuals

from above graph the normality assumption about the simple linear regression model error seem reasonable