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.11-4: Hypothesis Tests in Simple Linear Regression11-4.1 Use of t-TestsSuppose we wish to testH0: b1= b1,0H1: b1 b1,0An appropriate test statistic would be2xxST/ˆˆ20,10bbSec 11-4 Hypothesis Tests in Simple Linear Regression(11-6)Copyright © 2014 John Wiley & Sons, Inc. All rights reserved.11-4: Hypothesis Tests in Simple Linear Regression 11-4.1 Use of t-TestsWe would reject the null hypothesis if|t0| > ta/2,n - 2The test statistic could also be written as:3)ˆ(ˆˆ10,110bbbseTSec 11-4 Hypothesis Tests in Simple Linear RegressionCopyright © 2014 John Wiley & Sons, Inc. All rights reserved.11-4: Hypothesis Tests in Simple Linear Regression 11-4.1 Use of t-TestsSuppose we wish to testH0: b0= b0,0H1: b0 b0,0An appropriate test statistic would be4)ˆ(ˆ1ˆˆ00,00220,000bbbbbseSxnTxxSec 11-4 Hypothesis Tests in Simple Linear Regression(11-7)Copyright © 2014 John Wiley & Sons, Inc. All rights reserved.11-4: Hypothesis Tests in Simple Linear Regression 11-4.1 Use of t-TestsWe would reject the null hypothesis if|t0| > ta/2,n - 25Sec 11-4 Hypothesis Tests in Simple Linear RegressionCopyright © 2014 John Wiley & Sons, Inc. All rights reserved.11-4: Hypothesis Tests in Simple Linear Regression 11-4.1 Use of t-TestsAn important special case of the hypotheses of Equation 11-18 isH0: b1= 0H1: b1 0These hypotheses relate to the significance of regression.Failure to reject H0is equivalent to concluding that there is no linear relationship between x and Y.6Sec 11-4 Hypothesis Tests in Simple Linear RegressionCopyright © 2014 John Wiley & Sons, Inc. All rights reserved.11-4: Hypothesis Tests in Simple Linear Regression 7EXAMPLE 11-2 Oxygen Purity Tests of Coefficients We will test for significance of regression using the model for the oxygen purity data from Example 11-1. The hypotheses areH0: b1= 0H1: b1 0and we will use a = 0.01. From Example 11-1 and Table 11-2 we haveso the t-statistic in Equation 11-6 becomesPractical Interpretation: Since the reference value of t is t0.005,18= 2.88, the value of the test statistic is very far into the critical region, implying that H0: b1= 0 should be rejected. There is strong evidence to support this claim. The P-value for this test is . This was obtained manually with a calculator.Table 11-2 presents the Minitab output for this problem. Notice that the t-statistic value for the slope is computed as 11.35 and that the reported P-value is P = 0.000. Minitab also reports the t-statistic for testing the hypothesis H0: b0= 0. This statistic is computed from Equation 11-7, with b0,0= 0, as t0= 46.62. Clearly, then, the hypothesis that the intercept is zero is rejected.18.1ˆ,68088.0,20947.14ˆ21bxxSn35.11/0.6808818.1947.14)ˆ(ˆ/ˆˆ1210bbbseStxx9~1023.1PSec 11-4 Hypothesis Tests in Simple Linear RegressionCopyright © 2014 John Wiley & Sons, Inc. All rights reserved.11-4: Hypothesis Tests in Simple Linear Regression 11-4.2 Analysis of Variance Approach to Test Significance of RegressionThe analysis of variance identity is(11-8)Symbolically,SST= SSR+ SSE (11-9)8 niiiniiniiyyyyyy121212ˆˆSec 11-4 Hypothesis Tests in Simple Linear RegressionCopyright © 2014 John Wiley & Sons, Inc. All rights reserved.11-4: Hypothesis Tests in Simple Linear Regression 11-4.2 Analysis of Variance Approach to Test Significance of RegressionIf the null hypothesis, H0: b1= 0 is true, the statistic(11-10)follows the F1,n-2distribution and we would reject if f0> fa,1,n-2.9 ERERMSMSnSSSSF 2//10Sec 11-4 Hypothesis Tests in Simple Linear RegressionCopyright © 2014 John Wiley & Sons, Inc. All rights reserved.11-4: Hypothesis Tests in Simple Linear Regression 11-4.2 Analysis of Variance Approach to Test Significance of RegressionThe quantities, MSRand MSEare called mean squares.Analysis of variance table:Note that MSE= 10Source of VariationSum of SquaresDegrees of FreedomMean SquareF0Regression1 MSRMSR/MSEErrorn - 2 MSETotalSSTn - 1xyRSSS1ˆbxyTESSSSS1ˆb2ˆSec 11-4 Hypothesis Tests in Simple Linear RegressionCopyright © 2014 John Wiley & Sons, Inc. All rights reserved.11-4: Hypothesis Tests in Simple Linear Regression 11EXAMPLE 11-3 Oxygen Purity ANOVA We will use the analysis of variance approach to test for significance of regression using the oxygen purity data model from Example 11-1. Recall that , Sxy= 10.17744, and n = 20. The regression sum of squares isand the error sum of squares isSSE= SST- SSR= 173.38 - 152.13 = 21.25The analysis of variance for testing H0: b1= 0 is summarized in the Minitab output in Table 11-2. The test statistic is f0= MSR/MSE= 152.13/1.18 = 128.86, for which we find that the P-value is , so we conclude that b1is not zero.There are frequently minor differences in terminology among computer packages. For example, sometimes the regression sum of squares is called the “model” sum of squares, and the error sum of squares is called the “residual” sum of squares.947.14ˆ,38.1731bTSS13.15217744.10)947.14(ˆ1bxyRSSS9~1023.1PSec 11-4 Hypothesis Tests in Simple Linear RegressionCopyright © 2014 John Wiley & Sons, Inc. All rights reserved.11-5: Confidence Intervals 11-5.1 Confidence Intervals on the Slope and InterceptDefinition12Under the assumption that the observation are normally and independently distributed, a 100(1 - a)% confidence interval on the slope b1in simple linear regression is(11-11)Similarly, a 100(1 - a)% confidence interval on the intercept b0is(11-12)xxnxxnStSt22/2,1122/2,1ˆˆˆˆbbbaabbbaaxxnxxnSxntSxnt222/2,00222/2,01ˆˆ1ˆˆSec 11-5 Confidence IntervalsCopyright © 2014 John Wiley & Sons, Inc. All rights reserved.11-5: Confidence Intervals 13EXAMPLE 11-4 Oxygen Purity Confidence Interval on the Slope We will find a 95% confidence interval on the slope of the regression line using the data in Example 11-1. Recall that
View Full Document