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Algebra 1

3.3.2 Plotting and Analyzing Residuals

Algebra 13.3.2 Plotting and Analyzing Residuals

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Activity

Video still of oranges being weighed

Here is the data table for orange weights from a previous lesson. Use this information to answer the questions.

Number of oranges Weight in kilograms
3 1.027
4 1.162
5 1.502
6 1.617
7 1.761
8 2.115
9 2.233
10 2.569
1.

Use the graphing tool or technology outside the course. Create the scatter plot and calculate the best fit line using the Desmos tool below.

2.

What level of accuracy makes sense for the slope and intercept values? Be prepared to show your reasoning.

3.

What does the linear model estimate for how much the weight increases for of each additional box of oranges?

4.

Compare the weight of the actual box with 3 oranges in it to the estimated weight of the box with 3 oranges in it. Be prepared to show your reasoning.

5.

How many oranges are in the box when the linear model estimates the weight best? Be prepared to show your reasoning.

6.

How many oranges are in the box when the linear model estimates the weight least well? Be prepared to show your reasoning.

Use the following information to answer questions 7 and 8.

The difference between the actual value and the value estimated by a linear model is called the residual.
  • If the actual value is greater than the estimated value, the residual is positive.
  • If the actual value is less than the estimated value, the residual is negative.
7.

For the orange weight data set, what is the residual for the line of best fit when there are 3 oranges?

8.

On the same axes as the scatter plot, plot this residual at the point where x=3x=3 and yy has the value of the residual.

9.

Find the residuals for each of the other points in the scatter plot and graph them.

A blank graph with a horizontal x-axis labeled 1 to 11 and a vertical y-axis labeled -0.5 to 0.5. Both axes have arrows, and the grid lines form squares. No data points or additional markings are present.

10.

Which point on the scatter plot has the residual closest to zero? What does this mean about the weight of the box with that many oranges in it?

11.

How can you use the residuals to decide how well a line fits the data?

Video: Finding Residuals

Watch the following video to learn more about finding residuals.

Self Check

The equation for a line of best fit is y = 2.2 x + 3.4 . Find the residual for the point ( 3 , 9 ) .
  1. 14.2
  2. 1
  3. 1
  4. 2.4

Additional Resources

Finding Residuals

The gestation time for an animal is the typical duration between conception and birth. The longevity of an animal is the typical lifespan for that animal. The gestation times, in days, and longevities, in years, for 13 types of animals are shown in the table below.

Animal Gestation Time (Days) Longevity (Years)

Baboon

187

20

Black Bear

219

18

Beaver

105

5

Bison

285

15

Cat

63

12

Chimpanzee

23

20

Cow

284

15

Dog

61

12

Fox (Red)

52

7

Goat

151

8

Lion

100

15

Sheep

154

12

Wolf

63

5

The equation of a line of best fit is y=9.875+0.02039xy=9.875+0.02039x, where xx represents the gestational time, in days, and yy represents longevity, in years.

A scatter plot with red data points and a blue trendline showing a slight positive correlation between the variables on the x-axis (0 to 300) and y-axis (0 to 25).

A lion’s gestation time is 100 days, and its longevity is 15 years. What does this line of best fit predict the lion’s longevity to be?

Solution

The line of best fit is 𝑦=9.875+0.02039(100)=11.9𝑦=9.875+0.02039(100)=11.9

The residual would be: actual value – estimated value or 1511.9=3.11511.9=3.1.

The residual value is 3.1.

Try it

Try It: Finding Residuals

In the table above, a dog has a gestation time of 61 days and a longevity of 12 years.

Using the same equation, y=9.875+0.02039xy=9.875+0.02039x, what would the residual value be?

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