Question

Given are data for two variables, x and y.

xi	6	11	15	18	20
yi	7	7	13	21	30

(a)

Develop an estimated regression equation for these data. (Round your numerical values to two decimal places.)

ŷ =

(b)Compute the residuals. (Round your answers to two decimal places.)

xi	yi	Residuals
6	7
11	7
15	13
18	21
20	30

(c)Develop a plot of the residuals against the independent variable x.

a) A residual plot has 5 points plotted on it. The horizontal axis ranges from 0 to 25 and is labeled: x. The vertical axis ranges from −6 to 6 and is labeled: Residuals. There is a horizontal line that spans the graph at 0 on the vertical axis. 2 points lie above the line and 3 lie below it. The points start above the line at x = 6 in the upper left corner of the plot, then are plotted from left to right in a downward, diagonal direction until reaching a low below the line at x= 15. They then continue to be plotted in an upward diagonal direction until reaching a high above the line at x = 20. The maximum residual is located at x = 20.

b) A residual plot has 5 points plotted on it. The horizontal axis ranges from 0 to 25 and is labeled: x. The vertical axis ranges from −6 to 6 and is labeled: Residuals. There is a horizontal line that spans the graph at 0 on the vertical axis. 2 points lie above the line and 3 lie below it. The points start above the line at x = 6 in the upper left corner of the plot, then are plotted from left to right in a downward, diagonal direction until reaching a low below the line at x= 15. They then continue to be plotted in an upward diagonal direction until reaching a high above the line at x = 20. The maximum residual is located at x = 6.

c) A residual plot has 5 points plotted on it. The horizontal axis ranges from 0 to 25 and is labeled: x. The vertical axis ranges from −6 to 6 and is labeled: Residuals. There is a horizontal line that spans the graph at 0 on the vertical axis. 2 points lie above the line and 3 lie below it. The points are plotted from left to right in a downward, diagonal direction starting from the upper left corner of the plot. The maximum residual is located at x = 6 and the minimum occurs at x = 20.

d) A residual plot has 5 points plotted on it. The horizontal axis ranges from 0 to 25 and is labeled: x. The vertical axis ranges from −6 to 6 and is labeled: Residuals. There is a horizontal line that spans the graph at 0 on the vertical axis. 2 points lie above the line and 3 lie below it. The points are plotted from left to right in an upward, diagonal direction starting from the lower left corner of the plot. The minimum residual is located at x = 6 and the maximum occurs at x = 20.

Do the assumptions about the error terms seem to be satisfied?

a) The plot suggests curvature in the residuals indicating that the error term assumptions are satisfied.

b) The plot suggests a generally horizontal band of residual points indicating that the error term assumptions are not satisfied.  

c) The plot suggests a funnel pattern in the residuals indicating that the error term assumptions are not satisfied.

d) The plot suggests a generally horizontal band of residual points indicating that the error term assumptions are satisfied.

e) The plot suggests curvature in the residuals indicating that the error term assumptions are not satisfied.

(d)Compute the standardized residuals. (Round your answers to two decimal places.)

xi	yi	Standardized Residuals
6	7
11	7
15	13
18	21
20	30

(e) Develop a plot of the standardized residuals against ŷ.

a) A standardized residual plot has 5 points plotted on it. The horizontal axis ranges from 0 to 30 and is labeled: y hat. The vertical axis ranges from −3 to 3 and is labeled: Standardized Residuals. There is a horizontal line that spans the graph at 0 on the vertical axis. 2 points lie above the line and 3 lie below it. The points start above the line at about y hat = 3 in the upper left corner of the plot, then are plotted from left to right in a downward, diagonal direction until reaching a low below the line at about y hat = 17. They then continue to be plotted in an upward diagonal direction until reaching a high above the line at about y hat = 25.

b) A standardized residual plot has 5 points plotted on it. The horizontal axis ranges from 0 to 30 and is labeled: y hat. The vertical axis ranges from −3 to 3 and is labeled: Standardized Residuals. There is a horizontal line that spans the graph at 0 on the vertical axis. 3 points lie above the line and 2 lie below it. The points start below the line at about y hat = 3 in the lower left corner of the plot, then are plotted from left to right in an upward, diagonal direction until reaching a high above the line at about y hat = 17. They then continue to be plotted in a downward, diagonal direction until reaching a low below the line at about y hat = 25.

c) A standardized residual plot has 5 points plotted on it. The horizontal axis ranges from 0 to 30 and is labeled: y hat. The vertical axis ranges from −3 to 3 and is labeled: Standardized Residuals. There is a horizontal line that spans the graph at 0 on the vertical axis. 2 points lie above the line and 3 lie below it. The points are plotted from left to right in an upward, diagonal direction starting from the lower left corner of the plot. The points are between 2 to 26 on the horizontal axis and between −1 to 1.6 on the vertical axis.

d) A standardized residual plot has 5 points plotted on it. The horizontal axis ranges from 0 to 30 and is labeled: y hat. The vertical axis ranges from −3 to 3 and is labeled: Standardized Residuals. There is a horizontal line that spans the graph at 0 on the vertical axis. 2 points lie above the line and 3 lie below it. The points are plotted from left to right in a downward, diagonal direction starting from the upper left corner of the plot. The points are between 2 to 26 on the horizontal axis and between −1 to 1.6 on the vertical axis.

What conclusions can you draw from this plot?

a) The standardized residual plot has the same shape as the original residual plot. The plot suggests a generally horizontal band of residual points indicating that the error term assumptions are not satisfied.

b) The standardized residual plot has a different shape than the original residual plot. The plot suggests curvature in the residuals indicating that the error term assumptions are satisfied.    

c) The standardized residual plot has a different shape than the original residual plot. The plot suggests a generally horizontal band of residual points indicating that the error term assumptions are satisfied.

d) The standardized residual plot has a different shape than the original residual plot. The plot suggests a funnel pattern in the residuals indicating that the error term assumptions are not satisfied.

e) The standardized residual plot has the same shape as the original residual plot. The plot suggests curvature in the residuals indicating that the error term assumptions are not satisfied.

Answer 1

The given data is as follows:

X	Y
6	7
11	7
15	13
18	21
20	30

A simple linear regression equation is of the form Y= a +bX+e

Here,

• Y – Dependent variable
• X – Independent (explanatory) variable
• a – Intercept
• b – Slope
• e – Residual (error)

The value of the constants can be extimated using the following formulae:

and

From the above given values of variables X and Y we have:

and  

So, the estimated regression equation is:

The residual values are as follows:

Estimated y	e= Estimated Y -Real Y
2.90	4.10
10.84	-3.84
17.19	-4.19
21.95	-0.95
25.12	4.88

The plot is as follows: