Before we take up the discussion of linear regression and correlation, we need to examine a way to display the relation between two variables x and y. The most common and easiest way is a scatter plot. The following example illustrates a scatter plot.
Example 12.5
An educational researcher collects data on the vocabulary size of children as a function of age. The data is shown in Table 12.1. Is there a relationship between age and vocabulary size for young children? Construct a scatter plot. Let x = Child’s Age, and let y = Vocabulary Size.
| Age (years) | Vocabulary Size (number of words) |
|---|---|
| 3 | 655 |
| 4 | 1098 |
| 6 | 2463 |
| 7 | 3195 |
Using the TI-83, 83+, 84, 84+ Calculator
- Enter your X data into list L1 and your Y data into list L2.
- Press 2nd
STATPLOT ENTERto use Plot 1. On the input screen for PLOT 1, highlight On and pressENTER. (Make sure the other plots are OFF.) - For TYPE: highlight the very first icon, which is the scatter plot, and press
ENTER. - For Xlist:, enter
L1ENTER and for Ylist:L2ENTER. - For Mark: it does not matter which symbol you highlight, but the square is the easiest to see. Press
ENTER. - Make sure there are no other equations that could be plotted. Press Y = and clear any equations out.
- Press the
ZOOMkey and then the number 9 (for menu item "ZoomStat") ; the calculator will fit the window to the data. You can press WINDOW to see the scaling of the axes.
Try It 12.5
Amelia plays basketball for her high school. She wants to improve to play at the college level. She notices that the number of points she scores in a game goes up in response to the number of hours she practices her jump shot each week. She records the following data:
| X (hours practicing jump shot) | Y (points scored in a game) |
|---|---|
| 5 | 15 |
| 7 | 22 |
| 9 | 28 |
| 10 | 31 |
| 11 | 33 |
| 12 | 36 |
Construct a scatter plot and state if what Amelia thinks appears to be true.
A scatter plot shows the direction of a relationship between the variables. A clear direction happens when there is either:
- High values of one variable occurring with high values of the other variable or low values of one variable occurring with low values of the other variable.
- High values of one variable occurring with low values of the other variable.
You can determine the strength of the relationship by looking at the scatter plot and seeing how close the points are to a line, a power function, an exponential function, or to some other type of function. For a linear relationship there is an exception. Consider a scatter plot where all the points fall on a horizontal line providing a "perfect fit." The horizontal line would in fact show no relationship.
When you look at a scatterplot, you want to notice the overall pattern and any deviations from the pattern. The following scatterplot examples illustrate these concepts.
In this chapter, we are interested in scatter plots that show a linear pattern. Linear patterns are quite common. The linear relationship is strong if the points are close to a straight line, except in the case of a horizontal line where there is no relationship. If we think that the points show a linear relationship, we would like to draw a line on the scatter plot. This line can be calculated through a process called linear regression. However, we only calculate a regression line if one of the variables helps to explain or predict the other variable. If x is the independent variable and y the dependent variable, then we can use a regression line to predict y for a given value of x