In this chapter we will look relationships between two quantitative variables.

We now consider two quantitative variables, x and y, simultaneously. We will consider ordered pairs (x, y), and we will be interested in questions about what type, if any, of relationship occurs between them. We will consider the x as the explanatory or predictor variable and the y as the response variable. If there is a strong enough relationship between the two variables, knowing the value of x should allow us to predict a value of y.

Note that the association that exists between two variables is the same (positive, negative, strong, weak, or no real association) regardless of which variable is considered x and which is considered y. Also note, having an association between two variables (no matter how strong) does not mean there is a cause and effect relationship between them. There may be other factors that cause both variables to behave the way they do.

Discuss the type and strength of the association you would expect for the following pairs of variables. (a) (a city’s average temperature, amount of wood burned in fireplaces in that city) (b) (height, wingspan) (c) (height, shoe size) (d) (wingspan, # of siblings)

Using data from “ACSC”, the following scatterplots can be produced: (height, wingspan) (height, shoe size) (wingspan, # siblings)

Facts about r: values closer to -1 indicate strong negative correlation values closer to 1 indicate strong, positive correlation values closer to 0 indicate little or no correlation r has no units (regardless of the units of x and y) the value of r will be the same for (x, y) and (y, x)

Look at the scatterplot for the data given, discuss the association/correlation between the variables, and then calculate the value of r. x y537812

This process is tedious, but we can find the value of r using the TI 83/84. First we must put the data in the calculator as if we were creating a scatterplot. Then we press:

If we did this for the previous example (the one we just did by hand), we should see the screen below:

Find the correlation between the variables taken from data in “ACSC”. (a) (height, wingspan) (b) (height, shoesize) (c) (wingspan, # siblings)

Find the value of R 2 between the variables taken from data in “ACSC” and explain what each means. (a) (height, wingspan) (b) (height, shoesize) (c) (wingspan, # siblings)