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The P-value is P .0750 .0095 .0003 .0848. Thus, at the level 0.10, the null hypothesis is rejected and we conclude that the engineering changes have improved the process yield. This test procedure is sometimes called the Fisher-Irwin test. Because the test depends on the assumption that X1 X2 is xed at some value, some statisticians argue against use of the test when X1 X2 is not actually xed. Clearly X1 X2 is not xed by the sampling procedure in our example. However, because there are no other better competing procedures, the Fisher-Irwin test is often used whether or not X1 X2 is actually xed in advance.
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CHAPTER OUTLINE
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11-1 11-2 11-3 11-4 11-5 EMPIRICAL MODELS SIMPLE LINEAR REGRESSION PROPERTIES OF THE LEAST SQUARES ESTIMATORS SOME COMMENTS ON USES OF REGRESSION (CD ONLY) HYPOTHESIS TESTS IN SIMPLE LINEAR REGRESSION 11-5.1 Use of t-Tests 11-5.2 Analysis of Variance Approach to Test Signi cance of Regression 11-6 CONFIDENCE INTERVALS 11-6.1 Con dence Intervals on the Slope and Intercept 11-9 11-7 11-8
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Simple Linear Regression and Correlation
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11-6.2 Con dence Interval on the Mean Response PREDICTION OF NEW OBSERVATIONS ADEQUACY OF THE REGRESSION MODEL 11-8.1 Residual Analysis 11-8.2 Coef cient of Determination (R2)
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11-8.3 Lack-of-Fit Test (CD Only) TRANSFORMATIONS TO A STRAIGHT LINE
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11-10 MORE ABOUT TRANSFORMATIONS (CD ONLY) 11-11 CORRELATION
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LEARNING OBJECTIVES
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After careful study of this chapter, you should be able to do the following: 1. Use simple linear regression for building empirical models to engineering and scienti c 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 t to the data or to see if any underlying assumptions are violated 4. Test statistical hypotheses and construct con dence intervals on 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. Use simple transformations to achieve a linear regression model 7. Apply the correlation model
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11-1 EMPIRICAL MODELS
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CD MATERIAL 8. Conduct a lack-of- t test in a regression model where there are replicated observations. Answers for many odd numbered exercises are at the end of the book. Answers to exercises whose numbers are surrounded by a box can be accessed in the e-Text by clicking on the box. Complete worked solutions to certain exercises are also available in the e-Text. These are indicated in the Answers to Selected Exercises section by a box around the exercise number. Exercises are also available for some of the text sections that appear on CD only. These exercises may be found within the e-Text immediately following the section they accompany.
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EMPIRICAL MODELS
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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. This model can also be used for process optimization, such as nding the level of temperature that maximizes yield, or for process control purposes. As an illustration, consider the data in Table 11-1. In this table y is the purity of oxygen produced in a chemical distillation process, and x is the percentage of hydrocarbons that are present in the main condenser of the distillation unit. Figure 11-1 presents a scatter diagram
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Table 11-1 Oxygen and Hydrocarbon Levels Observation Number 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Hydrocarbon Level x (%) 0.99 1.02 1.15 1.29 1.46 1.36 0.87 1.23 1.55 1.40 1.19 1.15 0.98 1.01 1.11 1.20 1.26 1.32 1.43 0.95 Purity y (%) 90.01 89.05 91.43 93.74 96.73 94.45 87.59 91.77 99.42 93.65 93.54 92.52 90.56 89.54 89.85 90.39 93.25 93.41 94.98 87.33
100 98 96 Purity ( y) 94 92 90 88 86 0.85
1.15 1.25 1.35 Hydrocarbon level ( x)
Figure 11-1 Scatter diagram of oxygen purity versus hydrocarbon level from Table 11-1.