Question

A statistical program is recommended.

Consider the following data for two variables, x and y.

xi	135	110	130	145	175	160	120
yi	145	105	120	115	130	130	110

(a)

Compute the standardized residuals for these data. (Round your answers to two decimal places.)

xi	yi	Standardized Residuals
135	145
110	105
130	120
145	115
175	130
160	130
120	110

Do the data include any outliers? Explain. (Round your answers to two decimal places.)

The standardized residual with the largest absolute value is   , corresponding to y_i =   . Since this residual is  ---Select--- less than −2 between −2 and +2 greater than +2  , it  ---Select--- is definitely not could be  an outlier.

(b)

Plot the standardized residuals against ŷ.

A standardized residual plot has 7 points plotted on it. The horizontal axis ranges from 105 to 140 and is labeled: y hat. The vertical axis ranges from −2.5 to 2.5 and is labeled: Standardized Residuals. There is a horizontal line that spans the graph at 0 on the vertical axis. There are 4 points below the line and 3 points above it. 6 of the points appear to vary randomly between −0.8 to 0.1 on the vertical axis; however, the maximum residual is at approximately (121, 2.2).

A standardized residual plot has 7 points plotted on it. The horizontal axis ranges from 105 to 140 and is labeled: y hat. The vertical axis ranges from −2.5 to 2.5 and is labeled: Standardized Residuals. There is a horizontal line that spans the graph at 0 on the vertical axis. There are 3 points below the line and 4 points above it. 6 of the points appear to vary randomly between −0.1 to 0.8 on the vertical axis; however, the minimum residual is at approximately (121, −2.2).

A standardized residual plot has 7 points plotted on it. The horizontal axis ranges from 105 to 140 and is labeled: y hat. The vertical axis ranges from −2.5 to 2.5 and is labeled: Standardized Residuals. There is a horizontal line that spans the graph at 0 on the vertical axis. There are 4 points below the line and 3 points above it. The points are plotted from left to right in a downward, diagonal direction starting from the upper left corner of the graph. Most of the points are between −0.8 to 0.1 on the vertical axis; however, the maximum residual is at approximately (112, 2.2).

A standardized residual plot has 7 points plotted on it. The horizontal axis ranges from 105 to 140 and is labeled: y hat. The vertical axis ranges from −2.5 to 2.5 and is labeled: Standardized Residuals. There is a horizontal line that spans the graph at 0 on the vertical axis. There are 4 points below the line and 3 points above it. The points are plotted from left to right in an upward, diagonal direction starting from the lower left corner of the graph. Most of the points are between −0.8 to 0.1 on the vertical axis; however, the maximum residual is at approximately (134, 2.2).

Does this plot reveal any outliers?

The plot shows no possible outliers.The plot shows one possible outlier.    The plot shows two possible outliers.The plot shows more than two possible outliers.

(c)

Develop a scatter diagram for these data.

A scatter diagram has 7 points plotted on it. The horizontal axis ranges from 100 to 180 and is labeled: x. The vertical axis ranges from 90 to 150 and is labeled: y. The points are plotted from left to right in an upward, diagonal direction starting from the lower left corner of the diagram. The points are between 110 to 175 on the horizontal axis and between 105 to 145 on the vertical axis. Most of the points are plotted reasonably close together, but the fourth point from the left is noticeably higher than the others at 145 on the vertical axis.

A scatter diagram has 7 points plotted on it. The horizontal axis ranges from 100 to 180 and is labeled: x. The vertical axis ranges from 90 to 150 and is labeled: y. The points are plotted from left to right in a downward, diagonal direction starting from the upper left corner of the diagram. The points are between 110 to 175 on the horizontal axis and between 105 to 145 on the vertical axis. The points are fairly scattered, though the seventh point from left is slightly farther away from the others at 120 on the vertical axis.

A scatter diagram has 7 points plotted on it. The horizontal axis ranges from 100 to 180 and is labeled: x. The vertical axis ranges from 90 to 150 and is labeled: y. The points are plotted from left to right in a downward, diagonal direction starting from the upper left corner of the diagram. The points are between 110 to 175 on the horizontal axis and between 105 to 145 on the vertical axis. The points are fairly scattered, though the second point from the left is noticeably farther away from the others at 105 on the vertical axis.

A scatter diagram has 7 points plotted on it. The horizontal axis ranges from 100 to 180 and is labeled: x. The vertical axis ranges from 90 to 150 and is labeled: y. The points are plotted from left to right in an upward, diagonal direction starting from the lower left corner of the diagram. The points are between 110 to 175 on the horizontal axis and between 105 to 145 on the vertical axis. The points are reasonably close together and each consecutive point is higher than or just as high on the the diagram as the previous point.

Does the scatter diagram indicate any outliers in the data?

The diagram indicates that there are no possible outliers.The diagram indicates that there is one possible outlier.    The diagram indicates that there are two possible outliers.The diagram indicates that there are more than two possible outliers.

In general, what implications does this finding have for simple linear regression?

For simple linear regression, we must calculate standardized residuals, plot a standardized residual plot, and construct a scatter diagram to identify an outlier.For simple linear regression, we can determine an outlier by looking at the scatter diagram.     For simple linear regression, it is impossible to determine whether there is an outlier using standardized residuals, a standardized residual plot, or a scatter diagram.

Answer 1

(a)

Studentized
Residual
2.12
-0.74
0.09
-0.80
-0.54
0.07
-0.51

The standardized residual with the largest absolute value is 2.12, corresponding to y_i = 145 . Since this residual is greater than +2  , it could be  an outlier.

(b) The plot is:

The plot shows one possible outlier.   

(c) The plot is:

The diagram indicates that there is one possible outlier.   

For simple linear regression, we must calculate standardized residuals, plot a standardized residual plot, and construct a scatter diagram to identify an outlier.