Section 6.2 Prediction

Understanding Prediction 3 Key Principles of Prediction Prediction is based on fitting some “model” to a set of data. We are using a straight line model for our data. Prediction works best when the model fits the data closely. Prediction outside the range of data is risky. Imagine if we used a child growth chart to predict height at age 40!

The square of the regression, or r2, is a numerical quantity that tells us how well the least squares line predicts values of the response variable, y. r2 in regression The square of the correlation, r2, is the fraction of the variation in the values of y that is explained by the least-squares regression of y on x. The idea is that when there is a linear relationship, some of the variation in y is accounted for the fact that, as x changes, it pulls y along the regression line with it.

The coefficient of determination (denoted by r2) is a key output of regression analysis. It is interpreted as the proportion of the variance in the dependent variable that is predictable from the independent variable. The coefficient of determination is the square of the correlation (r) between predicted y scores and actual y scores; thus, it ranges from 0 to 1. With linear regression, the coefficient of determination is also equal to the square of the correlation between x and y scores. An r2 of 0 means that the dependent variable cannot be predicted from the independent variable. An r2 of 1 means the dependent variable can be predicted without error from the independent variable. An r2 between 0 and 1 indicates the extent to which the dependent variable is predictable. An r2 of 0.10 means that 10 percent of the variance in Y is predictable from X; an r2 of 0.20 means that 20 percent is predictable; and so on.

An r2 of 0 means that the dependent variable cannot be predicted from the independent variable. An r2 of 1 means the dependent variable can be predicted without error from the independent variable. An r2 between 0 and 1 indicates the extent to which the dependent variable is predictable. An r2 of 0.10 means that 10 percent of the variance in Y is predictable from X; an r2 of 0.20 means that 20 percent is predictable; and so on.

Enter the data into your calculator to find r and r2. Example: The table below lists data on body weight and backpack weight for 8 students. Body Weight 120 187 109 103 131 165 158 116 Backpack Weight 26 30 24 29 35 31 28 Enter the data into your calculator to find r and r2.

Because r2 = 0.632 for these data, about 63% of the observed variation in backpack weight can be explained by the straight line pattern What about the other 37%? It is due to other factors, such as physical fitness and motivation.

Exercises True or False: “When r = 0.7, this means that y can be predicted from x for 70% of the individuals in the sample.” False. It means that 49% of the variation in y can be explained by x. It says nothing about the individuals in the sample.

2. ) True or False: “When r2 = 0 2.) True or False: “When r2 = 0.7, this means that y can be predicted from x for 70% of the individuals in the sample.” False. While this statement used the correct numerical measurement for prediction, r2, it is still interpreting it incorrectly.

A researcher uses a regression equation to predict home heating bills (dollar cost), based on home size (square feet). The correlation between predicted bills and home size is 0.70. What is the correct interpretation of this finding? 70% of the variability in home heating bills can be explained by home size. (B) 49% of the variability in home heating bills can be explained by home size. (C) For each added square foot of home size, heating bills increased by 70 cents. (D) For each added square foot of home size, heating bills increased by 49 cents. (E) None of the above.

Complete the following exercises in the text: Page 377/6.29, 6.30, 6.31, 6.33