Lesson 21
The Slope of a Fitted Line
Let's look at how changing one variable changes another.
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=20240722T133821Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=f2740a446b6167e86a529742f86b6034aeab3dbb9a800bf22ae29e66f22f964a)
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.
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=20240722T133821Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=0f337f43a5f478913c48b512e5a8275c494981da4f18b1eae53cde7418c60d90)
Describe the relationship between the number of at bats and the number of hits using the data in the scatter plot.
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=20240722T133821Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=1fbf1f751544c3d4d53d7bf8547b00cf98f91a2fcbc34d0dc4c6b53fdc994c2d)
![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=20240722T133821Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=79218ae64e74d1e35c489b999357cdc30b82cb411f36dea6a11ee8b694fcfaa8)
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.
Problem 4
Solve: \(\begin{cases} y=\text-3x+13 \\ y=\text-2x+1 \\ \end{cases}\)
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=20240722T133821Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=9d8aa927e1674439644864387a0aba9022b839f3e31aa694f4ac8d7e4560c000)
- Circle any data that appear to be outliers.
- Use the graph to estimate the difference between any outliers and their predicted values.
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=20240722T133821Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=e92604af0a0d94a4689b2b72d6eb1d69f810a6a4819f82ee4e35fa966fe71420)