Lesson 21
The Slope of a Fitted Line
Problem 1
Which of these statements is true about the data in the scatter plot?
![Scatterplot. Horizontal, from 0 to 20, by 5’s. Vertical, from 0 to 60, by 15’s. 14 data points. Trend downward and to right.](https://cms-im.s3.amazonaws.com/MVnquqDQZq8H654ftYix6bZK?response-content-disposition=inline%3B%20filename%3D%228-8.6.PP.B.Image.23.png%22%3B%20filename%2A%3DUTF-8%27%278-8.6.PP.B.Image.23.png&response-content-type=image%2Fpng&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF3XOEFOW4%2F20240718%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240718T113847Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=00781124b4822b2da577bb9457f7832dae7439d3a96817e4a1b438033b513540)
As \(x\) increases, \(y\) tends to increase.
As \(x\) increases, \(y\) tends to decrease.
As \(x\) increases, \(y\) tends to stay unchanged.
\(x\) and \(y\) are unrelated.
Solution
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Problem 2
Here is a scatter plot that compares hits to at bats for players on a baseball team.
![Scatterplot, at bats, 0 to 600, hits, 0 to 150. Points begin at 10 comma 13 and trend up and to the right toward 590 comma 150.](https://cms-im.s3.amazonaws.com/QHReXMAHfJ22NcSxAr7KPoC6?response-content-disposition=inline%3B%20filename%3D%228-8.6.PP.B.Image.03.png%22%3B%20filename%2A%3DUTF-8%27%278-8.6.PP.B.Image.03.png&response-content-type=image%2Fpng&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF3XOEFOW4%2F20240718%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240718T113847Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=e2f497127e9167a40a51cf3d6229bb41d1ee069d38746b2212881342d1b8a72a)
Describe the relationship between the number of at bats and the number of hits using the data in the scatter plot.
Solution
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Problem 3
The linear model for some butterfly data is given by the equation \(y = 0.238x + 4.642\). Which of the following best describes the slope of the model?
![Photograph. Butterfly on a leaf.](https://cms-im.s3.amazonaws.com/FkA67LaPBixAtG1xE2yq6BEK?response-content-disposition=inline%3B%20filename%3D%228-8.5-butterfly.png%22%3B%20filename%2A%3DUTF-8%27%278-8.5-butterfly.png&response-content-type=image%2Fpng&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF3XOEFOW4%2F20240718%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240718T113847Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=7c7e16516989cbe896a1844626de294a2d8049c128c43e440e1e2803e5160085)
![Scatterplot of butterfly wingspan.](https://cms-im.s3.amazonaws.com/WoTFBsowhP8AzAHEERnPK66B?response-content-disposition=inline%3B%20filename%3D%228.6.PP.B.Image.28.png%22%3B%20filename%2A%3DUTF-8%27%278.6.PP.B.Image.28.png&response-content-type=image%2Fpng&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF3XOEFOW4%2F20240718%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240718T113847Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=c15e5e6ded19ea0e6713edae188868210cd5a59906e6e5eb1618ff768b6903f2)
For every 1 mm the wingspan increases, the length of the butterfly increases 0.238 mm.
For every 1 mm the wingspan increases, the length of the butterfly increases 4.642 mm.
For every 1 mm the length of the butterfly increases, the wingspan increases 0.238 mm.
For every 1 mm the length of the butterfly increases, the wingspan increases 4.642 mm.
Solution
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Problem 4
Solve: \(\begin{cases} y=\text-3x+13 \\ y=\text-2x+1 \\ \end{cases}\)
Solution
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(From Unit 5, Lesson 15.)Problem 5
Nonstop, one-way flight times from O’Hare Airport in Chicago and prices of a one-way ticket are shown in the scatter plot.
![Scatterplot.](https://cms-im.s3.amazonaws.com/muFpyAczSnhkhuTkJX4aa4yP?response-content-disposition=inline%3B%20filename%3D%228-8.6.PP.B.Image.11.png%22%3B%20filename%2A%3DUTF-8%27%278-8.6.PP.B.Image.11.png&response-content-type=image%2Fpng&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF3XOEFOW4%2F20240718%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240718T113847Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=fc140f94a1e338a781af45aac47c7a7a3bce24bf994d3bbf8a20d76f0ec43121)
- Circle any data that appear to be outliers.
- Use the graph to estimate the difference between any outliers and their predicted values.
Solution
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(From Unit 5, Lesson 19.)Problem 6
Consider the following graphs of linear equations. Decide which line has a positive slope, and which has a negative slope. Then calculate each line’s exact slope.
![Graph of two lines, l and m, origin O, with grid.](https://cms-im.s3.amazonaws.com/nyZSu8R7Kve9Xzsk4bYYktYe?response-content-disposition=inline%3B%20filename%3D%228-8.3.PP.01.png%22%3B%20filename%2A%3DUTF-8%27%278-8.3.PP.01.png&response-content-type=image%2Fpng&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF3XOEFOW4%2F20240718%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240718T113847Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=3d7ee17791403fbf56ae8b8223c76b236dd59d31efda2cdce7ea453aaa6cbdce)
Solution
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(From Unit 5, Lesson 9.)