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.11-7: Adequacy of the Regression Model • Fitting a regression model requires several assumptions.1. Errors are uncorrelated random variables with mean zero;2. Errors have constant variance; and,3. Errors be normally distributed.• The analyst should always consider the validity of these assumptions to be doubtful and conduct analyses to examine the adequacy of the model2Sec 11-7 Adequacy of the Regression ModelCopyright © 2014 John Wiley & Sons, Inc. All rights reserved.11-7: Adequacy of the Regression Model 11-7.1 Residual Analysis• The residuals from a regression model are ei= yi- ŷi, where yiis an actual observation and ŷiis the corresponding fitted value from the regression model. • Analysis of the residuals is frequently helpful in checking the assumption that the errors are approximately normally distributed with constant variance, and in determining whether additional terms in the model would be useful.3Sec 11-7 Adequacy of the Regression ModelCopyright © 2014 John Wiley & Sons, Inc. All rights reserved.11-7: Adequacy of the Regression Model 4EXAMPLE 11-7 Oxygen Purity ResidualsThe regression model for the oxygen purity data in Example 11-1 is . Table 11-4 presents the observed and predicted values of y at each value of x from this data set, along with the corresponding residual. These values were computed using Minitab and show the number of decimal places typical of computer output. A normal probability plot of the residuals is shown in Fig. 11-10. Since the residuals fall approximately along a straight line in the figure, we conclude that there is no severe departure from normality. The residuals are also plotted against the predicted value in Fig. 11-11 and against the hydrocarbon levels xiin Fig. 11-12. These plots do not indicate any serious model inadequacies.ˆ74.283 14.947yxiyˆSec 11-7 Adequacy of the Regression ModelCopyright © 2014 John Wiley & Sons, Inc. All rights reserved.11-7: Adequacy of the Regression Model 5Sec 11-7 Adequacy of the Regression ModelExample 11-7Copyright © 2014 John Wiley & Sons, Inc. All rights reserved.11-7: Adequacy of the Regression Model Example 11-7Figure 11-10 Normal probability plot of residuals, Example 11-7. 6Sec 11-7 Adequacy of the Regression ModelCopyright © 2014 John Wiley & Sons, Inc. All rights reserved.11-7: Adequacy of the Regression Model Example 11-7Figure 11-11 Plot of residuals versus predicted oxygen purity, ŷ, Example 11-7. 7Sec 11-7 Adequacy of the Regression ModelCopyright © 2014 John Wiley & Sons, Inc. All rights reserved.11-7: Adequacy of the Regression Model 11-7.2 Coefficient of Determination (R2)• The quantityis called the coefficient of determination and is often used to judge the adequacy of a regression model.• 0 R2 1;• We often refer (loosely) to R2as the amount of variability in the data explained or accounted for by the regression model.8TETRSSSSSSSSR 12Sec 11-7 Adequacy of the Regression ModelCopyright © 2014 John Wiley & Sons, Inc. All rights reserved.11-7: Adequacy of the Regression Model 11-7.2 Coefficient of Determination (R2)• For the oxygen purity regression model, R2= SSR/SST= 152.13/173.38 = 0.877• Thus, the model accounts for 87.7% of the variability in the data.9Sec 11-7 Adequacy of the Regression ModelCopyright © 2014 John Wiley & Sons, Inc. All rights reserved.11-8: Correlation 10We assume that the joint distribution of Xiand Yiis the bivariate normal distribution presented in Chapter 5, and mYand are the mean and variance of Y, mXand are the mean and variance X, and r is the correlation coefficient between Y and X. Recall that the correlation coefficient is defined as(11-15)where sXYis the covariance between Y and X.The conditional distribution of Y for a given value of X = x is(11-16)where(11-17)(11-18)2Ys2XsYXXYsssr ss2|10||21exp21xYxYxYxyyfXYXYssrmm0rssXY1Sec 11-8 CorrelationCopyright © 2014 John Wiley & Sons, Inc. All rights reserved.11-8: Correlation We may also write:11It is possible to draw inferences about the correlation coefficient r in this model. The estimator of r is the sample correlation coefficient(11-19)Note that(11-20) /212/112121TXXXYniiniiniiiSSSSYYXXXXYR RSSSXXT/211ˆTRTYXYYXXSSSSSSSSSR 1212ˆˆSec 11-8 CorrelationCopyright © 2014 John Wiley & Sons, Inc. All rights reserved.11-8: Correlation It is often useful to test the hypothesesH0: r = 0H1: r 0The appropriate test statistic for these hypotheses isReject H0if |t0| > t/2,n-2.122012RnRT(11-21)Sec 11-8 CorrelationCopyright © 2014 John Wiley & Sons, Inc. All rights reserved.11-8: Correlation The test procedure for the hypothesisH0: r = 0H1: r 0where r0 0 is somewhat more complicated. In this case, the appropriate test statistic is Reject H0if |z0| > z/2.13Z0= (arctanh R - arctanh r0)(n - 3)1/2(11-22)Sec 11-8 CorrelationCopyright © 2014 John Wiley & Sons, Inc. All rights reserved.11-8: Correlation The approximate 100(1- )% confidence interval is14/2 /2tanh arctanh tanh arctanh33zzrrnn r (11-23)Sec 11-8 CorrelationCopyright © 2014 John Wiley & Sons, Inc. All rights reserved.11-8: Correlation 15EXAMPLE 11-8 Wire Bond Pull Strength In Chapter 1 (Section 1-3) an application of regression analysis is described in which an engineer at a semiconductor assembly plant is investigating the relationship between pull strength of a wire bond and two factors: wire length and die height. In this example, we will consider only one of the factors, the wire length. A random sample of 25 units is selected and tested, and the wire bond pull strength and wire length arc observed for each unit. The data are shown in Table 1-2. We assume that pull strength and wire length are jointly normally distributed.Figure 11-13 shows a scatter diagram of wire bond
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