1. Using the regression line to make predictions about Y

Suppose you are interested in the relationship between foster care and incarceration for young adults who have been involved in the juvenile justice system. You are able to access some anonymous data from 75 30-year-old graduates of the juvenile justice system. The following table shows the data from four of these graduates from your sample. Their results are typical of the rest of your sample. You would like to use the number of foster placements to make predictions about the number of arrests.

Table: Number of Foster Placements and Number of Arrests

Graduate	Foster Placements	Arrests
Tina	4	7
Matt	2	4
Nikki	1	4
Tyler	3	5

You can use the preceding sample data to obtain the least-squares regression line:

Y=a+bX

To be able to use this line to predict Y values based on a known value of X, you will first have to calculate the slope and the Y-intercept. The first step in calculating these values is to calculate the means of X and Y.

The mean X score is X = , and the mean Y score is Y = .

Now you are ready to complete the following computation table:

Tina	4		7
Matt	2		4
Nikki	1		4
Tyler	3		5

To calculate the slope and the Y-intercept, you first will have to calculate the covariation of X and Y, (XX)(YY)

, and the sum of the squared deviations around the mean of X, (XX)2

. Luckily, you have already done most of the work.

The covariation of X and Y is , and the sum of the squared deviations around the mean of X is .

Now you can calculate the slope and the Y-intercept.

The slope of the least-squares regression line is .

The Y-intercept of the least-squares regression line is .

You now have everything you need to use the least-squares regression line to estimate the number of arrests for someone who had five foster placements. The least-squares regression line suggests that a 30-year-old graduate of the juvenile justice system who had five foster placements would have arrests.

What might the implications of this research finding be on policy?