13- Chapter Thirteen McGraw- © 2005 The McGraw-Hill Companies, Inc., All Chapter Thirteen 13- Linear Regression and Correlation GOALS When you have completed this chapter, you will be able to: ONE Draw a scatter diagram TWO Understand and interpret the terms dependent variable and independent variable THREE Calculate and interpret the coefficient of correlation, the coefficient of determination, and the standard error of estimate FOUR Conduct a test of hypothesis to determine if the population coefficient of correlation is different from zero Goals Chapter Thirteen 13- continued Linear Regression and GOALSCorrelation When you have completed this chapter, you will be able to: FIVE Calculate the least squares regression line and interpret the slope and intercept values SIX Construct and interpret a confidence interval and prediction interval for the dependent variable SEVEN Set up and interpret an ANOVA table Goals 13- Correlation Analysis is a group of statistical techniques to measure the association between two variables A Scatter Diagram is a chart that portrays the relationship between two variables The Dependent Variable is the variable being predicted or estimated The Independent Variable provides the basis for estimation It is the predictor variable Correlation Analysis 13- The Coefficient of Correlation (r) is a measure of the strength of the relationship between two variables Also called Pearson’s r and It requires interval or ratioPearson’s product moment scaled data correlation coefficient P e a r s o n 's r It can range from -1.00 to 1.00 Values of -1.00 or 1.00 indicate perfect and strong -1 correlation Negative values indicate an Values close to 0.0 indicate inverse relationship and weak correlation positive values indicate a The Coefficient of Correlation, r direct relationship 13- Y 10 0 X 10 Perfect Negative Correlation 13- Y 10 0 X 10 Perfect Positive Correlation 13- Y 10 0 X 10 Zero Correlation 13- Y 10 0 X 10 Strong Positive Correlation 13- 10 We calculate the coefficient of correlation from the following formula r= Σ(X – X)(Y – Y) (n-1)sxsy Formula for r The regression equation is Y’= a + bX 13- 22 where Y’ is the average predicted value of Y for any X a is the Y-intercept It is the estimated Y value when X=0 b is the slope of the line, or the average change in Y’ for each change of one unit in X The least squares principle is used to obtain a and b Regression Analysis 13- 23 The least squares principle is used to obtain a and b The equations to determine a and b are: sy b=r sx a = Y – bX Regression Analysis 13- 24 Develop a regression equation for the information given in example that can be used to estimate the selling price based on the number of pages sy b=r sx = (.657) 12.21 138.87 = 0578 a = Y – bX = 79.5 - 0578*625 = 43.39 Example revisited 13- 25 The regression equation is: Y’ = 43.39 + 0578X The slope of the line is 0578 Each addition page costs about a nickel The sign of the b value and the sign of r will always be the same The equation crosses the Y-axis at $43.39 A book with no pages would cost $43.39 Example revisited 13- 26 We can use the regression equation to estimate values of Y The estimated selling price of an 800 page book is $89.61, found by Price = $43.39 + 0578(Number of Pages) = $43.39 + 0578(800) = $89.61 Example revisited 13- 27 The Standard Error of Estimate measures the scatter, or dispersion, of the observed values around the line of regression The formula that is used to compute the standard error: sy x = √ Σ ( Y - Y ') n -2 The Standard Error of Estimate 13- 28 Find the standard error of estimate for the problem involving the number of pages in a book and the selling price Estimated Actual price price Deviation (Y) (Y') (Y-Y') 84 72.28 11.72 75 83.83 -8.83 99 89.61 9.39 72 78.06 -6.06 69 66.50 2.50 81 78.06 2.94 63 78.06 -15.06 93 89.61 3.39 0.00 Deviation Squared (Y-Y') 137.41 78.03 88.15 36.67 6.25 8.67 226.67 11.48 593.33 sy x = √ Σ ( Y - Y ') n -2 = √ 3 -2 = 9.944 Example revisited 13- 29 Assumptions Underlying Linear Regression For each value of X, there is a group of Y values, and these Y values are normally distributed The Y values are statistically independent This means that in the selection of a sample, the Y values chosen for a particular X value not depend on the Y values for any other X values The means of these normal distributions of Y values all lie on the straight line of regression The standard deviations of these normal distributions are the same 13- 30 The confidence interval for the mean value of Y for a given value of X is given by: Y ' + t(s y x) √ + (X -X )2 n Σ (X -X )2 Y’is the predicted value for any selected X value X is an selected value of X X is the mean of the Xs n is the number of observations Sy.x is the standard error of the estimate t is the value of t at n-2 degrees of freedom Confidence Interval 13- 31 For our earlier price estimate of $89.61, the confidence interval, assuming a desired 95% confidence, is calculated as follows Example Y’ the predicted value, is $89.61 13- 32 X is 800 pages X is 625, the mean of the pages n is 8, the number of observations Sy.x is 9.944, the standard error of the estimate t is 2.447 at 8-2 degrees of freedom and 95% confidence + 9 4 (2 4 ) √ (8 0 - )2 + ,0 0 = + 4 3 Y ' + t(s y x) √ + (X -X )2 n Σ (X -X )2 Example revisited 13- 33 The prediction interval for an individual value of Y for a given value of X Y ' + t(s y x) √ + + 9 4 (2 4 ) = + + (X -X )2 n Σ (X -X )2 √ (8 0 - )2 + + ,0 0 Prediction Interval 13- 34 Summarizing The Results The estimated selling price for a book with 800 pages is $89.61 The standard error of estimate is $9.94 The 95 percent confidence interval for all books with 800 pages is $89.61 + $14.43 This means the limits are between $75.18 and $104.04 The 95 percent prediction interval for a particular book with 800 pages is $89.61+ $28.29 The means the limits are between $61.32 and $117.90 These results appear in the following Minitab and Excel outputs Example revisited 13- 35 Regression Analysis The regression equation is Price = 43.4 + 0.0578 No of Pages Predictor Coef StDev T P Constant 43.39 17.28 2.51 0.046 No of Pages 0.05778 0.02706 2.13 0.077 S = 9.944 R-Sq = 43.2% R-Sq(adj) = 33.7% Analysis of Variance Source DF SS MS F P Regression 450.67 450.67 4.56 0.077 Error 593.33 98.89 Total 1044.00 Fit StDev Fit 95.0% CI 95.0% PI 89.61 5.90 ( 75.17, 104.05) ( 61.31, 117.91) Example revisited M I N I T A B 13- 36 Regression Statistics Multiple R 0.657 R Square 0.432 Adjusted R Square 0.337 Standard Error 9.944 Observations ANOVA df SS MS F Significance F Regression 450.67 450.67 4.5573034 0.0767 Residual 593.33 98.89 Total 1044 Intercept Page Coefficients 43.3889 0.0578 Standard Error 17.277 0.027 t Stat 2.511 2.135 E X C E L P-value 0.0458193 0.0767009 Example revisited ... and interpret a confidence interval and prediction interval for the dependent variable SEVEN Set up and interpret an ANOVA table Goals 13- Correlation Analysis is a group of statistical techniques. .. 13 Book Page Price($) Introduction to History 500 84 Basic Algebra 700 75 Introduction to Psychology 800 99 Introduction to Sociology 600 72 Business Management 400 69 Introduction to Biology... the Y-intercept It is the estimated Y value when X=0 b is the slope of the line, or the average change in Y’ for each change of one unit in X The least squares principle is used to obtain a and