3.3 Bivariate descriptive statistics: linear regression and

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.

From an article in the Wall Street Journal : In Europe and Asia, m-commerce is popular. M-commerce users have special mobilephones 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 amachine 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

A scatter plot shows the direction and strength 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.

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.

Positive linear pattern (strong)

Linear pattern w/ one deviation

Negative linear pattern (strong)

Negative linear pattern (weak)

Exponential growth pattern

No pattern

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.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 orpredict 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 $x$ .

Before we calculate the regression equation there are some condition we should think about first; quantitative data condition, straight enough condition, and outlier condition. The quantitative data condition is making sure that you are working with two numerical variables. Remember that sometimes we use number to represent categories. The straight enough condition is making sure the data is linear. Look at the scatterplot, do the data look linear? Lastly, the outlier condition is looking to see if you have data points on the scatterplot that do not seem to fit the overall trend in the data. If you identify points that you think are outliers you may want to remove them from the dataset and calculate the regression equation again without them in the dataset.