49 12.3 The Regression Equation
Data rarely fit a straight line exactly. Usually, you must be satisfied with rough predictions. Typically, you have a set of data whose scatter plot appears to “fit” a straight line. This is called a Line of Best Fit or Least-Squares Line.
Example 1
A random sample of 11 statistics students produced the following data, where x is the third exam score out of 80, and y is the final exam score out of 200. Can you predict the final exam score of a random student if you know the third exam score?
| x (third exam score) | y (final exam score) |
|---|---|
| 65 | 175 |
| 67 | 133 |
| 71 | 185 |
| 71 | 163 |
| 66 | 126 |
| 75 | 198 |
| 67 | 153 |
| 70 | 163 |
| 71 | 159 |
| 69 | 151 |
| 69 | 159 |
Table showing the scores on the final exam based on scores from the third exam.
Scatter plot showing the scores on the final exam based on scores from the third exam.
Try It
SCUBA divers have maximum dive times they cannot exceed when going to different depths. The data in the table show different depths with the maximum dive times in minutes.
| Depth (in feet) | Maximum dive time (in minutes) |
|---|---|
| 50 | 80 |
| 60 | 55 |
| 70 | 45 |
| 80 | 35 |
| 90 | 25 |
| 100 | 22 |
- Can you predict the maximum dive time of a random diver if you know the depth?
- What is the maximum dive time if a diver dives at 110 feet?
Show Answer
- max dive time = 127.24 – 1.1143 * depth
- At 110 feet, a diver could dive for only five minutes.
The third exam score, x, is the independent variable and the final exam score, y, is the dependent variable. We will plot a regression line that best “fits” the data. If each of you were to fit a line “by eye,” you would draw different lines. We can use what is called a least-squares regression line to obtain the best fit line.
Consider the following diagram. Each point of data is of the the form (x, y) and each point of the line of best fit using least-squares linear regression has the form [latex]\displaystyle{({x}\hat{{y}})}[/latex].
The [latex]\displaystyle\hat{{y}}[/latex] is read “y hat” and is the estimated value of y. It is the value of y obtained using the regression line. It is not generally equal to y from data.
The term [latex]\displaystyle{y}_{0}-\hat{y}_{0}={\epsilon}_{0}[/latex] is called the “error” or residual. It is not an error in the sense of a mistake. The absolute value of a residual measures the vertical distance between the actual value of y and the estimated value of y. In other words, it measures the vertical distance between the actual data point and the predicted point on the line.
|
In the diagram above, [latex]\displaystyle{y}_{0}-\hat{y}_{0}={\epsilon}_{0}[/latex] is the residual for the point shown. Here the point lies above the line and the residual is positive.
ε = the Greek letter epsilon
For each data point, you can calculate the residuals or errors,
[latex]{\epsilon}_{i} = {y}_{i}-\hat{y}_{i}[/latex] for i = 1, 2, 3, …, 11.
Each |ε| is a vertical distance.
For the example