1. Least Squares Regression
4.2
Least Squares Regression

2. Best Fit
If we have two variables X and Y, we often would like to model the relation as a line
Draw a line through the scatter diagram
We want to find the line that “best” describes the linear relationship … the regression line

3. Linear Model We want to use a linear model
Linear models can be written in several different (equivalent) ways
y = m x + b
y – y1 = m (x – x1)
y = b1 x + b0
Because the slope and the intercept both are important to analyze, we will use

4. Residual = Observed – Predicted
Residuals
One difference between math and stat is that statistics assumes that the measurements are not exact, that there is an error or residual
The formula for the residual is always
Residual = Observed – Predicted
This relationship is not just for this chapter … it is the general way of defining error in statistics

5. Residual What the residual is on the scatter diagram The residual
The model line
The observed value y
The predicted value y
The x value of interest

6. Least-Squares Regression Line
We want to minimize the residuals, but we need to define what this means
We use the method of least-squares
We consider a possible linear mode
We calculate the residual for each point
We add up the squares of the residuals
The line that has the smallest is called the least-squares regression line

7. Least Squares Regression Line
The equation for the least-squares regression line is given by
y = b1x + b0
b1 is the slope of the least-squares regression line
b0 is the y-intercept of the least-squares regression line

8. Tough Stuff
Finding the values of b1 and b0, by hand, is a very tedious process
You should use calculator for this
Finding the coefficients b1 and b0 is only the first step of a regression analysis
We need to interpret the slope b1
We need to interpret the y-intercept b0

9. Slope Interpreting the slope b1
The slope is sometimes defined as
The slope is also sometimes defined as
The slope relates changes in y to changes in x

10. Y-intercept Where the graph crosses the y-axis
Interpreted as the location where “x” = 0
Think about what is means in the situation for the “x” to be zero

11. © 2010 Pearson Prentice Hall. All rights reserved
If the least-squares regression line is used to make predictions based on values of the explanatory variable that are much larger or much smaller than the observed values, we say the researcher is working outside the scope of the model. Never use a least-squares regression line to make predictions outside the scope of the model because we can’t be sure the linear relation continues to exist.
© 2010 Pearson Prentice Hall. All rights reserved
4-11

12. Example
The speed of a golf club and the distance the ball went were measured for a linear relationship. Draw a scatter pot of the data, and the least-squares regression line. Write down the equation.

13. Data
Club Head Speed (mph)
Distance (yards)
100
257
102
264
103
274
101
266
105
277
263 99
258
275
Using your equation, estimate how far a golf ball would travel if it was hit at 104 mph.
Interpret the slope of the regression equation

14. Sample Problem
The heights and weights of 11 men between the ages of 21 and 26 were measured. The data are presented in the table below.
A. Create at a scatter diagram to confirm that an approximately linear relationship exists between x and y.
B. Find the least squares regression line, treating height, x, as the explanatory variable and weight, y, as the response variable
C. Interpret the slope
D. Interpret the y-intercept, is this appropriate.
d. Use the regression line to predict the weight of a man who is 73 inches tall?