1. Regression-Prediction The regression-prediction equations are the optimal linear equations for predicting Y from X or X from Y

2. Regression-Prediction Equations Equations predicting Y from X and X from Y:

3. Solving a Prediction Problem Identify X, Y, Means, Standard Deviations, and correlation coefficient, r XY. Identify X, Y, Means, Standard Deviations, and correlation coefficient, r XY. Write out Prediction Equation Write out Prediction Equation Substitute z-score definitions Substitute z-score definitions Substitute known values Substitute known values Solve for unknown Solve for unknown

4. Example Joe and his sister, Jane, were raised together in the same home. Both are now adults. If Jane’s IQ is 130, what do you guess that Joe’s IQ is? Joe and his sister, Jane, were raised together in the same home. Both are now adults. If Jane’s IQ is 130, what do you guess that Joe’s IQ is? The correlation in IQ for siblings raised together is.50. You can also assume that the mean IQ for both men and women is 100 and standard deviations are 15. The correlation in IQ for siblings raised together is.50. You can also assume that the mean IQ for both men and women is 100 and standard deviations are 15.

5. Label X, Y, and what is known. Let X = sister’s IQ Let X = sister’s IQ Let Y = brother’s IQ Let Y = brother’s IQ Jane’s IQ is given (130), and we are to predict her brother’s IQ. This means:

7. Summary of Solution The best prediction of Joe’s IQ is 115. The best prediction of Joe’s IQ is 115. Our prediction was based on Jane’s IQ. If we knew nothing about Joe, we would have made a guess of 100 (the mean). Our prediction was based on Jane’s IQ. If we knew nothing about Joe, we would have made a guess of 100 (the mean). Given that Jane’s IQ is high (130), we guess that Joe’s is also above average. Given that Jane’s IQ is high (130), we guess that Joe’s is also above average. However, the predicted value for Joe shows regression to the mean--only 115. However, the predicted value for Joe shows regression to the mean--only 115.

8. Next Topic: How accurate? The correlation was.50, which is why Joe’s IQ is predicted to be half-way between Jane’s IQ (130) and the mean (100). The correlation was.50, which is why Joe’s IQ is predicted to be half-way between Jane’s IQ (130) and the mean (100). How accurate is this prediction? The squared correlation is.25; so, if we use this equation, the sum of squared deviations is 25% less than if we always guess the mean. How accurate is this prediction? The squared correlation is.25; so, if we use this equation, the sum of squared deviations is 25% less than if we always guess the mean. In the next section, you will learn more about the accuracy of predictions. In the next section, you will learn more about the accuracy of predictions.