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2.03 Residuals

Worksheet
Residuals
1

The following tables show sets of data \left(x, y\right) and the predicted \hat{y} values based on a least squares regression line. Complete the tables by finding the residuals:

a
x13579
y22.722.324.221.821.5
\hat{y}25.223.421.619.818
\text{Residuals}
b
x56789
y37.737.221.127.144
\hat{y}28.930.431.933.434.9
\text{Residuals}
2

Complete the following tables of residuals:

a
xy\hat{y}\text{Residual}
102925
1437-8
7204
6138
205855
1326-8
b
xy\hat{y}\text{Residual}
12.8-33.30.6
11.4-30.3-30.3
11.1-29.70.2
6.5-21.5-1.4
10-28.5-27.4-1.1
3

If a residual is a positive value, is the actual value of the response variable above or below the least squares regression line?

4

If a measured point in a data set is below the least squares regression line, will the corresponding residual be positive or negative?

Residual plots
5

Construct the residual scatter plot of the following data sets:

a
5
10
15
x
-20
-15
-10
-5
y
b
5
10
x
-20
-10
10
20
30
40
y
6

The residual plot for a set of data is shown. Draw the scatter plot showing the original data.

1
2
3
4
5
6
7
8
9
10
x
-5
-4
-3
-2
-1
1
2
3
4
5
y
7

The residual plot can help us decide if a linear model should be used for a raw set of data.

a

Explain when a linear model would be a good fit for a set of raw data.

b

Explain when a linear model would not be a good fit for a set of raw data.

8

For each of the following data sets:

i
Complete the table of residuals.
ii

Plot the residuals on a scatter plot.

iii

Determine if the model is a good fit for the data.

a

The table shows a company's costs, C (in millions), in week W.

The equation C = 3 W + 4 is being used to model the data.

WC\text{Model value}\text{Residual}
17
29
521
726
1034
1343
1548
1858
b

The table shows a company's revenue, R (in millions), in week W.

The equation R = 3 W + 5 is being used to model the data.

WR\text{Model value}\text{Residual}
27
312
519
729
937
1038
1241
1445
9

The table below shows the residual values after a least-squares regression line has been fitted to a set of data:

a

Plot the residuals on a scatter plot.

b

Determine if a linear model is a good fit for the data.

x12201018920
\text{Residual}-4-2523-1
10

For each of the data sets below:

i

Complete the table by finding the residuals.

ii

Plot the residuals on a scatter plot.

iii

Determine whether the original data has a linear or nonlinear relationship.

iv

Create a scatter plot for the original data.

a
xy\hat{y}\text{Residuals}
214.512
4911
61210
879
1010.58
b
xyy\text{Residuals}
2308
42010
61412
81214
101416
122018
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Outcomes

ACMGM058

use a residual plot to assess the appropriateness of fitting a linear model to the data

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