Section 2.1 Linear Regression
Subsection 1. Read a scatterplot
We read the coordinates of points on a scatterplot the same way we do for any other graph.
Subsubsection Example
Example 2.1.
The scatterplot shows the height and shoe size of a group of men.
- State the height and shoe size of the man represented by point \(A\text{.}\)
- Find the heights of two men with the same shoe size.
![scatterplot](images/fig-1-6-1.png)
- The man represented by point \(A\) has shoe size \(11 \frac{1}{2}\) and is \(73 \frac{1}{2}\) inches tall.
- There are two men with shoe size 9, with heights 68 and 73 inches. There are also two men with shoe size \(9 \frac{1}{2}\text{,}\) with heights 68 and 71 inches.
Subsubsection Exercise
Checkpoint 2.2.
The scatterplot shows the heights of dance partners in a ballroom dance class.
- How tall is the shortest woman?
- What are the heights of the three partners of the 65-inch tall women?
- 59 in
- \(68 \frac{1}{2}\text{,}\) \(70\text{,}\) and \(71\) in
Subsection 2. Sketch a line of best fit
Of course, the points on a scatterplot may not lie on a straight line. But if they seem to cluster near a line, we can try to find that line.
Subsubsection Example
Example 2.3.
Sketch a line of best fit for the scatterplot in part 1.
We draw a line that lies as close as possible to all of the data points. As a rule of thumb, we try to keep equal numbers of points on each side of the line.
![scatterplot](images/fig-1-6-3.png)
Subsubsection Exercise
Checkpoint 2.4.
Which of the lines fits the scatterplot best?
![scatterplot](images/fig-1-6-4.png)
Subsection 3. Fit a line through two points
If we don't know the slope of a line, but we do know two points on the line, we can calculate the slope first and then use the point-slope formula.
Subsubsection Example
Example 2.5.
Find an equation for the line that passes through \((2,-1)\) and \((-1,3)\text{.}\)
We solve this problem in two steps: First, find the slope of the line, and then use the point-slope formula.
Step 1: Let \((x_1,y_1)=(2,-1)\) and \((x_2,y_2)=(-1,3)\text{.}\) Use the slope formula to find
Step 2: Apply the point-slope formula with \(m=\dfrac{-4}{3}\) and \((x_1,y_1)=(2,-1)\text{.}\) (We can use either point in the formula.) Then
Cross-multiply to find
Subsubsection Exercise
Checkpoint 2.6.
Find an equation for the line that passes through \((-6,-1)\) and \((1,3)\text{.}\)
Step 1: Compute the slope.
Step 2: Use the point-slope formula.