Plotting and Analyzing Residuals

3.3.2 Plotting and Analyzing Residuals

3.3.2 • Plotting and Analyzing Residuals

Activity

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

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

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Compare your answer:

The equation of the best fit line is $y = 0.216 x + 0.345$ .

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

Compare your answer:

Rounding the values to the thousandths place makes sense because that would be to an accuracy of 1 gram, which is what the original data used.

What does the linear model estimate for how much the weight increases for each additional orange that is placed in the box of oranges?

Compare your answer:

0.216 kg for each orange

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.

Compare your answer:

The actual weight is 1.027 kg, and the estimated weight is 0.993 kg. Therefore, the actual weight is 0.034 kg greater than the estimate.

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

Compare your answer:

6 oranges are in the box when the estimate is best. The difference between the estimated weight and the actual weight is only 0.023 kg, while the other points are farther from the line.

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

Compare your answer:

7 oranges are in the box when the estimate is worst. The difference between the estimated weight and the actual weight is 0.095, which is the most of any of the values.

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.

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

Compare your answer:

The residual is $0.034$ (because $1.027 - 0.993 = 0.034$ ).

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

Compare your answer:

The point is (8,0.043).

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.

Compare your answer:

A scatter plot with points at (2, 0.08), (3, 0.07), (5, 0.09), (6, 0.08), (7, 0.08), and (8, 0.07) on a grid. The x-axis ranges from 0 to 11 and the y-axis ranges from -0.1 to 0.5.

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?

Compare your answer: Your answer may vary, but here is a sample.

6 oranges. It means the estimated weight of the box with 6 oranges in it is closest to the actual value.

11.

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

Compare your answer: Your answer may vary, but here is a sample.

A line will result in more accurate estimates when the residuals are closer to zero.

Video: Finding Residuals

Watch the following video to learn more about finding residuals.

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Self Check

The equation for a line of best fit is

y = 2.2x +3.4

. Find the residual for the point

(3, 9)

$14.2$
$-1$
$1$
$2.4$

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 $y = 9.875 + 0.02039 x$ , where $x$ represents the gestational time, in days, and $y$ 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 $𝑦 = 9.875 + 0.02039 (100) = 11.9$

The residual would be: actual value – estimated value or $15 - 11.9 = 3.1$ .

The residual value is 3.1.

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, $y = 9.875 + 0.02039 x$ , what would the residual value be?

Compare your answer:

Substitute $x = 61$ into the equation, $𝑦 = 9.875 + 0.02039 (61) = 11.1$ .

So, the residual is the (actual value – estimated value):

$12 - 11.1 = 0.9$