Least Squares Regression 4.2 Least Squares Regression

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

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

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

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

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

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

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

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

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

© 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

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.

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

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?