Analyzing TwoVariable Data Lesson 2 7 Assessing a

Analyzing Two-Variable Data Lesson 2. 7 Assessing a Regression Model Statistics and Probability with Applications, 3 rd Edition Starnes & Tabor Bedford Freeman Worth Publishers

Assessing a Regression Model Learning Targets After this lesson, you should be able to: ü Use a residual plot to determine if a regression model is appropriate. ü Interpret the standard deviation of the residuals. ü Interpret r 2. Statistics and Probability with Applications, 3 rd Edition 2

Assessing a Regression Model Now that we have learned how to calculate a least-squares regression line, it is important to assess how well the line fits the data. We do this by asking two questions: • Is a line the right model to use, or would a curve be better? • If a line is the right model to use, how well does it make predictions? We can use residuals to assess whether a regression model is appropriate by making a residual plot. Residual Plot A residual plot is a scatterplot that plots the residuals on the vertical axis and the explanatory variable on the horizontal axis. Statistics and Probability with Applications, 3 rd Edition 3

Residual plot n n A scatterplot of the (x, residual) pairs. Residuals can be graphed against other statistics besides x Purpose is to tell if a linear association exist between the x & y variables If no pattern exists between the points in the residual plot, then the association is linear Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Linear Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Not linear

Range of Motion 35 154 24 142 40 137 31 133 28 122 25 126 26 135 14 108 20 120 21 127 30 122 One measure of the success of knee surgery is post-surgical range of motion for the knee joint following a knee dislocation. Is there a linear relationship between age & range of motion? Sketch a residual plot. Enter the data, run Lin Reg and then find RESID in list menu Residuals Age x Since there is no pattern in the residual plot, there is a linear relationship between age and range of motion Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Assessing a Regression Model Here is a scatterplot showing the relationship between Super Bowl number and the cost of a 30 -second commercial for the years 1967– 2013, along with the least-squares regression line. The resulting residual plot is also shown to the right of the scatterplot. The least-squares regression line clearly doesn’t fit this association very well! In the early years, the actual cost of an ad is always greater than the line predicts, resulting in positive residuals. Statistics and Probability with Applications, 3 rd Edition 7

Assessing a Regression Model Here is a scatterplot showing the Ford F-150 data from Lesson 2. 5, along with the corresponding residual plot. Looking at the scatterplot, the line seems to be a good fit for the association. You can “see” that the line is appropriate by the lack of a leftover pattern in the residual plot. In fact, the residuals look randomly scattered around the residual = 0 line. Statistics and Probability with Applications, 3 rd Edition 8

Assessing a Regression Model Interpreting a Residual Plot To determine if the regression model is appropriate, look at the residual plot. • If there is no leftover pattern in the residual plot, the regression model is appropriate. • If there is a leftover pattern in the residual plot, the regression model is not appropriate. Statistics and Probability with Applications, 3 rd Edition 9

Assessing a Regression Model Once we have all the residuals, we can measure how well the line makes predictions with the standard deviation of the residuals. Standard deviation of the residuals s The standard deviation of the residuals s measures the size of a typical residual. That is, s measures the typical distance between the actual y values and the predicted y values. To calculate the standard deviation of the residuals s, we square each of the residuals, add them, divide the sum by n – 2, and take the square root. Statistics and Probability with Applications, 3 rd Edition 10

Assessing a Regression Model Besides the standard deviation of the residuals s, we can also use the coefficient of determination r 2  to measure how well the regression line makes predictions. Coefficient of Determination r 2 The coefficient of determination r 2 measures the percent reduction in the sum of squared residuals when using the leastsquares regression line to make predictions rather than the mean value of y. In other words, r 2 measures the percent of the variability in the response variable that is accounted for by the least-squares regression line. Simply put, the coefficient of determination, r 2 , tells us that the LSRL improves our prediction for y by ______% Statistics and Probability with Applications, 3 rd Edition 11

Assessing a Regression Model The sum of squared residuals has been reduced by 66%. That is, 66% of the variability in the price of a Ford F-150 is accounted for by the least-squares regression line with x = miles driven. The remaining 34% is due to other factors, including age, color, condition, and other features of the truck. Statistics and Probability with Applications, 3 rd Edition 12

LESSON APP 2. 7 Do higher priced tablets have better battery life? Can you predict the battery life of a tablet using the price? Using data from a sample of 15 tablets, the least-squares regression line y^ = 4. 67 + 0. 0068 x was calculated using x = price (in dollars) and y = battery life (in hours). A residual plot for this model is shown. 1. Use the residual plot to determine whether the regression model is appropriate. 2. Interpret the value s =1. 21 for this model. 3. Interpret the value r 2 = 0. 342 for this model. Statistics and Probability with Applications, 3 rd Edition 13

Assessing a Regression Model Learning Targets After this lesson, you should be able to: ü Use a residual plot to determine if a regression model is appropriate. ü Interpret the standard deviation of the residuals. ü Interpret r 2. Statistics and Probability with Applications, 3 rd Edition 14