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

In Europe and Asia, m-commerce is popular. M-commerce users have special mobile phones that work like electronic wallets as well as provide phone and Internet services. Users can do everything from paying for parking to buying a TV set or soda from a machine to banking to checking sports scores on the Internet. For the years 2000 through 2004, was
there a relationship between the year and the number of m-commerce users? Construct a scatter plot. Let *x* = the year and let *y* = the number of m-commerce users, in millions.

$x$ (year) | $y$ (# of users) |
---|---|

2000 | 0.5 |

2002 | 20.0 |

2003 | 33.0 |

2004 | 47.0 |

### 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 ENTER to use Plot 1. On the input screen for PLOT 1, highlight On and press ENTER. (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 L1 ENTER and for Ylist: L2 ENTER.
- 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 ZOOM key 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*