Chapter 13 - Linear regression and correlation, after studying this chapter you will be able to: Identify a relationship between variables on a scatter diagram, measure and interpret a degree of relationship by a coefficient of correlation, conduct a test of hypothesis about the coefficient of correlation in a population,...and other contents.
13 1 Copyright © 2004 by The McGrawHill Companies, Inc. All rights reserved. 13 2 13 2 When you have completed this chapter, you will be able to: Identify a relationship between variables on a scatter diagram Measure and interpret a degree of relationship by a coefficient of correlation Conduct a test of hypothesis about the coefficient of correlation in a population Identify the roles of dependent and independent variables, the concept of regression, and its distinction from the concept of correlation Copyright © 2004 by The McGrawHill Companies, Inc. All rights reserved. 13 3 13 3 Measure and interpret the strength of relationship between two variables through a regression line and the technique of least squares Conduct analysis of variance and calculate coefficient of determination Conduct a test of hypothesis for a regression model and each coefficient of regression Estimate confidence and prediction intervals Copyright © 2004 by The McGrawHill Companies, Inc. All rights reserved. Terminology 13 4 Correlation Analysis …is a group of statistical techniques used to measure the strength of the association between two variables Scatter Diagram …is a chart that portrays the relationship between the two variables. Dependent Variable …is the variable being predicted or estimated Independent Variable …provides the basis for estimation. It is the predictor variable Copyright © 2004 by The McGrawHill Companies, Inc. All rights reserved. The Coefficient of Correlation…rr The Coefficient of Correlation… 13 5 … Is a measure of strength of the relationship between two variables … It requires interval or ratioscaled data … It can range from 1.00 to 1.00 …Values of 1.00 or 1.00 indicate perfect and strong correlation …Values close to 0.0 indicate weak correlation …Negative values indicate an inverse relationship and positive values indicate a direct relationship Copyright © 2004 by The McGrawHill Companies, Inc. All rights reserved. Perfect Negative Negative Correlation Correlation Perfect Y 10 9 8 7 6 5 4 3 2 1 0 0 1 2 3 4 5 6 7 8 9 10 X Copyright © 2004 by The McGrawHill Companies, Inc. All rights reserved. 13 6 Perfect Positive Positive Correlation Correlation Perfect Y 10 9 8 7 6 5 4 3 2 1 0 0 1 2 3 4 5 6 7 8 9 10 X Copyright © 2004 by The McGrawHill Companies, Inc. All rights reserved. 13 7 Zero Correlation Correlation Zero Y 10 9 8 7 6 5 4 3 2 1 0 0 1 2 3 4 5 6 7 8 9 10 X Copyright © 2004 by The McGrawHill Companies, Inc. All rights reserved. 13 8 Strong Positive Positive Correlation Correlation Strong Y 13 9 10 9 8 7 6 5 4 3 2 1 0 0 1 2 3 4 5 6 7 8 9 10 X Copyright © 2004 by The McGrawHill Companies, Inc. All rights reserved. Example Example 13 10 Chart 136 Copyright © 2004 by The McGrawHill Companies, Inc. All rights reserved. Using Excel 13 42 ls o o T n o k c i l C See See Click on Click on DATA ANALYSIS DATA ANALYSIS Copyright © 2004 by The McGrawHill Companies, Inc. All rights reserved. See See Using Excel 13 43 …Click OK Highlight REGRESSION Copyright © 2004 by The McGrawHill Companies, Inc. All rights reserved. See See Using Excel INPUT NEEDS INPUT NEEDS Copyright © 2004 by The McGrawHill Companies, Inc. All rights reserved. 13 44 …Click OK See See Using Excel Copyright © 2004 by The McGrawHill Companies, Inc. All rights reserved. 13 45 See See Using Excel The regression equation The regression equation is: is: y = 0.07x +22.6 y = 0.07x +22.6 Copyright © 2004 by The McGrawHill Companies, Inc. All rights reserved. 13 46 The Standard The Standard Error of Error of Estimate Estimate 13 47 …this measures the scatter, or dispersion, of the observed values around the line of regression The formulas that are used to compute the standard error are: Se (y y ) n y Copyright © 2004 by The McGrawHill Companies, Inc. All rights reserved. 2 a y b xy n The Standard The Standard Error of Error of 13 48 Estimate Estimate Find the standard error of estimate for the problem involving the number of pages in a book and the selling price Previously: Previously: Se x y xy x2 y2 4 900 636 397 200 3 150 000 51 606 y a y b xy n 51,606 48(636) 0.05143(397,200) 10.408 10.408 Copyright © 2004 by The McGrawHill Companies, Inc. All rights reserved. Assumptions Underlying Assumptions Underlying 13 49 Linear Linear Regression Regression For each value of x, there is a group of y values, and these y values are normally distributed 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 equal The y values are statistically independent. This means that in the selection of a sample the y values chosen for a particular x value do not depend on the y values for any other x values Copyright © 2004 by The McGrawHill Companies, Inc. All rights reserved. 13 50 Confidence Interval x y xy x2 y2 Previously: Previously: 4 900 636 397 200 3 150 000 51 606 The confidence interval for the mean value of y for a given value of x is given by: 89 14 447 ( 10 408 ) 15 31 Copyright © 2004 by The McGrawHill Companies, Inc. All rights reserved. x 89 14 ( x2 ) ( x) n x y0 tα/2(n2) Se n ( 800 612 ) ( 4 900 ) 150 000 13 51 Prediction Interval x y xy x2 y2 Previously: Previously: 4 900 636 397 200 3 150 000 51 606 The prediction interval for an individual value of y for a given value of x is given by: 447 ( 10 408 ) 89 14 29 72 Copyright © 2004 by The McGrawHill Companies, Inc. All rights reserved. 1 + x 89 14 ( x2 ) ( x) n x y0 tα/2(n2) Se n ( 800 612 ) ( 4 900 ) 150 000 13 52 Summarizing the Summarizing the Results Results The estimated selling price for a book with 800 pages is $89.14 The standard error of estimate is $10.41 The 95 percent confidence interval for all books with 800 pages is $89.14 + $15.31 This means the limits are between $73.83 and $104.45 The 95 percent prediction interval for a particular book with 800 pages is $89.14 + $29.72 The means the limits are between $59.42 and These results appear in the following MINITAB output These results appear in the following MINITAB output $118.86 Copyright © 2004 by The McGrawHill Companies, Inc. All rights reserved. 13 53 Regression Analysis: Price versus Pages The regression equation is Price = 48.0 + 0.0514 Pages Predictor Constant Pages Coef 48.00 0.05143 S = 10.41 SE Coef 16.94 0.02700 R-Sq = 37.7% T 2.83 1.90 P 0.030 0.105 R-Sq(adj) = 27.3% Analysis of Variance Source Regression Residual Error Total DF SS 393.4 650.6 1044.0 MS 393.4 108.4 F 3.63 Predicted Values for New Observations P 0.105 New Obs Fit 89.14 SE Fit 6.26 95.0% CI (73.82,104.46) Copyright © 2004 by The McGrawHill Companies, Inc. All rights reserved. 95.0% PI (59.41,118.88) EXCEL output: EXCEL output: Price vs. Pages Price vs. Pages Copyright © 2004 by The McGrawHill Companies, Inc. All rights reserved. 13 54 Test your learning… … Test your learning … … n o n o k ilcick CCl www.mcgrawhill.ca/college/lind Online Learning Centre for quizzes extra content data sets searchable glossary access to Statistics Canada’s EStat data …and much more! Copyright © 2004 by The McGrawHill Companies, Inc. All rights reserved. 13 55 13 56 This completes Chapter 13 Copyright © 2004 by The McGrawHill Companies, Inc. All rights reserved. ... coefficient of correlation in a population Identify the roles of dependent and independent variables, the concept of regression, and its distinction from the concept of correlation Copyright © 2004 by The McGrawHill Companies, Inc. All rights reserved. ... …Values of 1.00 or 1.00 indicate perfect and strong correlation …Values close to 0.0 indicate weak correlation …Negative values indicate an inverse relationship and positive values indicate a direct relationship... Conduct analysis of variance and calculate coefficient of determination Conduct a test of hypothesis for a regression model and each coefficient of regression Estimate confidence and prediction intervals