4.2

Least Squares Regression

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

Best Fit

● 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

y = b1 x + b0

Linear Model

● 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 alwaysResidual = Observed – Predicted

● This relationship is not just for this chapter … it is the general way of defining error in statistics

Residuals

What the residual is on the scatter diagram

Residual

The model line

The x value of interest

The observed value y

The residual

The predicted value y

● 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 smallestis called the least-squares regression line

Least-Squares Regression Line

The equation for the least-squares

regression line is given byy = b1x + b0

b1 is the slope of the least-squares regression line

b0 is the y-intercept of the least-squares regression line

Least Squares Regression Line

● 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

Tough Stuff

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

Slope

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

Y-intercept

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.

4-11

© 2010 Pearson Prentice Hall. All rights reserved

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.

Example

Club Head Speed (mph)

Distance (yards)

100 257

102 264

103 274

101 266

105 277

100 263

99 258

105 275

Data

• 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

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?