### Introduction

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) | $y$ (no. 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 first icon, which is the scatter plot, then press`ENTER`

. - For Xlist, enter
`L1 ENTER`

; for Ylist, enter`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 out any equations. - 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.

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, or
- 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 (see figures above). When you look at a scatter plot, you want to notice the overall pattern and any deviations from the pattern.

In this chapter, we are interested in scatter plots that show a linear pattern. Linear patterns are common. The linear relationship is strong if the points are close to a straight line. If we think the points show a linear relationship, we draw a line on the scatter plot. This line can be calculated through a process called *linear regression*. A linear regression line models the trend of the data. However, we only calculate a regression line if one of the variables helps explain or predict the other variable. If *x* is the independent variable and *y* is the dependent variable, then we can use a regression line to predict *y* for a given value of *x.*