Copyright © 2014 John Wiley & Sons, Inc. All rights reserved.Copyright © 2014 John Wiley & Sons, Inc. All rights reserved.Chapter 11Simple Linear Regression and CorrelationApplied Statistics and Probability for EngineersSixth EditionDouglas C. Montgomery George C. RungerCopyright © 2014 John Wiley & Sons, Inc. All rights reserved.211Simple Linear Regression and Correlation11-1 Empirical Models11-2 Simple Linear Regression11-3 Properties of the Least Squares Estimators11-4 Hypothesis Test in Simple Linear Regression11-4.1 Use of t-tests11-4.2 Analysis of variance approach to test significance of regression11-5 Confidence Intervals11-5.1 Confidence intervals on the slope and intercept11-5.2 Confidence interval on the mean response11-6 Prediction of New Observations11-7 Adequacy of the Regression Model11-7.1 Residual analysis11-7.2 Coefficient of determination (R2)11-8 Correlation11-9 Regression on Transformed Variables11-10 Logistic RegressionCHAPTER OUTLINEChapter 11 Title and OutlineCopyright © 2014 John Wiley & Sons, Inc. All rights reserved.Learning Objectives for Chapter 11After careful study of this chapter, you should be able to dothe following:1. Use simple linear regression for building empirical models to engineering and scientific data.2. Understand how the method of least squares is used to estimate the parameters in a linear regression model.3. Analyze residuals to determine if the regression model is an adequate fit to the data or to see if any underlying assumptions are violated.4. Test the statistical hypotheses and construct confidence intervals on the regression model parameters.5. Use the regression model to make a prediction of a future observation and construct an appropriate prediction interval on the future observation.6. Apply the correlation model.7. Use simple transformations to achieve a linear regression model.3Chapter 11 Learning ObjectivesCopyright © 2014 John Wiley & Sons, Inc. All rights reserved.11-1: Empirical Models• Many problems in engineering and science involve exploring the relationships between two or more variables. • Regression analysis is a statistical technique that is very useful for these types of problems. • For example, in a chemical process, suppose that the yield of the product is related to the process-operating temperature. • Regression analysis can be used to build a model to predict yield at a given temperature level.4Sec 11-1 Empirical ModelsCopyright © 2014 John Wiley & Sons, Inc. All rights reserved.11-2: Simple Linear Regression • The simple linear regression considers a single regressor or predictor x and a dependent or response variable Y.• The expected value of Y at each level of x is a random variable:E(Y|x) = b0+ b1x• We assume that each observation, Y, can be described by the modelY = b0+ b1x + 5Sec 11-2 Simple Linear RegressionCopyright © 2014 John Wiley & Sons, Inc. All rights reserved.11-2: Simple Linear Regression Least Squares Estimates6The least-squares estimates of the intercept and slope in the simple linear regression model are(11-1)(11-2)where and .xybbˆˆbniniiiniiniiniiinxxnxyxy1212111ˆiniyny1)/1(inixnx1)/1(Sec 11-2 Simple Linear RegressionCopyright © 2014 John Wiley & Sons, Inc. All rights reserved.11-2: Simple Linear Regression 7The fitted or estimated regression line is therefore(11-3)Note that each pair of observations satisfies the relationshipwhere is called the residual. The residual describes the error in the fit of the model to the ithobservation yi. xybbˆˆˆniexyiii,,2,1,ˆˆbbiiiyyeˆSec 11-2 Simple Linear RegressionCopyright © 2014 John Wiley & Sons, Inc. All rights reserved.11-2: Simple Linear Regression Notation8 nxxxxSniiniiniixx211212 nyxyxxxySniiniiniiiniiixy11112Sec 11-2 Simple Linear RegressionCopyright © 2014 John Wiley & Sons, Inc. All rights reserved.EXAMPLE 11-1 Oxygen Purity9We will fit a simple linear regression model to the oxygen purity data in Table 11-1. The following quantities may be computed:6566.214,22892.295321.044,1701605.921960.121.843,192.232020120122012201201iiiiiiiiiiiyxxyyxyxn68088.020)92.23(2892.2920222012012iiiixxxxS17744.1020)21.843,1()92.23(6566.214,220201201201iiiiiiixyyxyxSandSec 11-2 Simple Linear RegressionCopyright © 2014 John Wiley & Sons, Inc. All rights reserved.EXAMPLE 11-1 Oxygen Purity - continued10Therefore, the least squares estimates of the slope and intercept areandThe fitted simple linear regression model (with the coefficients reported to three decimal places) is94748.1468088.017744.10ˆ1bxxxySS283 31.74196.1)94748.14(160 5.92ˆˆ0bbxyxy 947.14283.74ˆSec 11-2 Simple Linear RegressionCopyright © 2014 John Wiley & Sons, Inc. All rights reserved.11-2: Simple Linear Regression Estimating 2The error sum of squares isIt can be shown that the expected value of the error sum of squares is E(SSE) = (n – 2)2.11 niiiniiEyyeSS1212ˆSec 11-2 Simple Linear RegressionCopyright © 2014 John Wiley & Sons, Inc. All rights reserved.11-2: Simple Linear Regression Estimating 2An unbiased estimator of 2 iswhere SSEcan be easily computed using122ˆ2nSSE(11-4)xyTESSSSS1ˆb(11-5)Sec 11-2 Simple Linear RegressionCopyright © 2014 John Wiley & Sons, Inc. All rights reserved.11-3: Properties of the Least Squares Estimators • Slope Properties• Intercept Properties1311)ˆ( bbExxSV21)ˆ(bbbbxxSxnVE220001)ˆ(and)ˆ(Sec 11-3 Properties of the Least Squares
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