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://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=20240718T032710Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=e4d073f1497f0a0e2fde4d71c257475e83392962fb6e31ca68df355ffa5fbe0c)
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://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=20240718T032710Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=7c677f786ebb9f2f33437d13e3ec0002f4881602e73f67ffc027c65845dec67a)
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://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=20240718T032710Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=12b6abf4f60fa5f6ba16544fc7e34ea3ae0923ee6ce0b1d1d9583314bf0b44b2)
![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=20240718T032710Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=44a82aae490316e90b5ce62328e8d1fe018e069bd8fd351e02d92c930781c385)
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://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=20240718T032710Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=6d37b706a345624765b283033b476f3db8033407ee10ce86bb6f21f34c1041be)
- 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://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=20240718T032710Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=c64128c5aa4486e96254d951a162ea0a60789eb5eaf284d08a15687384b750ee)