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://staging-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=AKIAXQCCIHWF37H2AMFB%2F20240722%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240722T142810Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=b664257c461a07b8325bbfa60f357aca7df72095ba022bf17f96d7895e35dae8)
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://staging-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=AKIAXQCCIHWF37H2AMFB%2F20240722%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240722T142810Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=6d11acc10b3471b71f5fb51a6f08ecc519d37400e3df74528b3174e637aa7e82)
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://staging-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=AKIAXQCCIHWF37H2AMFB%2F20240722%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240722T142810Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=5b738c86053f07f333afcaa5b76342dfeb352d3dae88ec8060f3b1353cbb3ba9)
![Scatterplot of butterfly wingspan.](https://staging-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=AKIAXQCCIHWF37H2AMFB%2F20240722%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240722T142810Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=053470c83461b3d0f96d8acf3de59783d64b3c4caca765d0c6cdabe581af6219)
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://staging-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=AKIAXQCCIHWF37H2AMFB%2F20240722%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240722T142810Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=382c62c55710bb3d65c936946a024e7437c857cdc1ca9be53701f11cd018632e)
- 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://staging-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=AKIAXQCCIHWF37H2AMFB%2F20240722%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240722T142810Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=60ba2de4b01d6cacfe872a3780fb4e240a699a9b4b061df5eaf7595d82aec806)
Solution
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(From Unit 5, Lesson 9.)