9.3 Modeling Linear Relationships
If you know a person’s pinky (smallest) finger length, do you think you could predict that person’s height? Imagine collecting data on this and constructing a scatterplot of the points on graph paper. Then draw a line that appears to “fit” the data. For your line, pick two convenient points and use them to find the slope of the line. Find the y-intercept of the line by extending your line so it crosses the y-axis. Using the slopes and the y-intercepts, write your equation of “best fit.” According to your equation, what is the predicted height for a pinky length of 2.5 inches? You have just started the process of linear regression.
Linear Regression
Data rarely perfectly fit a straight line, but we can 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. This process of fitting the best-fit line is called linear regression.
The equation of the regression line is ŷ =a+bx
The ŷ is read “y hat” and is the estimated value of y. It is the value of y obtained using the regression line. It may or may not be equal to values of y observed from the data.
The sample means of the x values and the y values are and , respectively. The best fit line always passes through the point .
The slope, b can be written as where sy = the standard deviation of the y values and sx = the standard deviation of the x values. r is the correlation coefficient, which is discussed in the next section.
The y-intercept, a, can then be calculated by using the slope, and means of x and y.
Example
Recall our example:
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 |
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 (x, ŷ).
Your turn!
SCUBA divers have maximum dive times they cannot exceed when going to different depths. The data in the figure below show different depths with the maximum dive times in minutes. Use your calculator to find the least squares regression line and predict the maximum dive time for 110 feet.
| X (depth in feet) | Y (maximum dive time) |
|---|---|
| 50 | 80 |
| 60 | 55 |
| 70 | 45 |
| 80 | 35 |
| 90 | 25 |
| 100 | 22 |
Understanding Slope
The slope of the line, b, describes how changes in the variables are related. It is important to interpret the slope of the line in the context of the situation represented by the data. You should be able to write a sentence interpreting the slope in plain Englis