Transcript Regression-Prediction - California State University, Fullerton

Regression-Prediction
The regression-prediction equations
are the optimal linear equations for
predicting Y from X or X from Y
Regression-Prediction Equations
Equations predicting Y from X and X from Y:
Yˆi  Y
zˆYi 
 rXY z X i
sY
Xˆ i  X
zˆX i 
 rXY zYi
sX
Solving a Prediction Problem
Identify X, Y, Means, Standard Deviations,
and correlation coefficient, rXY.
 Write out Prediction Equation
 Substitute z-score definitions
 Substitute known values
 Solve for unknown

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?
 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.

Label X, Y, and what is known.
Let X = sister’s IQ
 Let Y = brother’s IQ

X  Y 100;sX  sY 15;rXY  .50.
Jane’s IQ is given (130), and we are to predict
her brother’s IQ. This means: X  130
i
Yˆi  ?
X  Y  100;sX  sY  15;rXY  .50.
zˆYi  rXY zX i
Yˆi  Y
Xi  X
 rXY
sY
sX
130100
Yˆi 100
 .50

 15

15
Yˆ  115
i
Summary of Solution
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).
 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.

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).
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.