Applied Statistics and Probability for Engineers

11-1 EMPIRICAL MODELS 373

CD MATERIAL

8. Conduct a lack-of-fit 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.

11-1 EMPIRICAL MODELS

Many problems in engineering and science involve exploring the relationships between two

or more variables. Regression analysisis 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 finding 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 yis the purity of oxygen

produced in a chemical distillation process, and xis the percentage of hydrocarbons that are

present in the main condenser of the distillation unit. Figure 11-1 presents a scatter diagram

Table 11-1 Oxygen and Hydrocarbon Levels

Observation Hydrocarbon Level Purity

Number x(%) y(%)

1 0.99 90.01

2 1.02 89.05

3 1.15 91.43

4 1.29 93.74

5 1.46 96.73

6 1.36 94.45

7 0.87 87.59

8 1.23 91.77

9 1.55 99.42

10 1.40 93.65

11 1.19 93.54

12 1.15 92.52

13 0.98 90.56

14 1.01 89.54

15 1.11 89.85

16 1.20 90.39

17 1.26 93.25

18 1.32 93.41

19 1.43 94.98

20 0.95 87.33 Figure 11-1 Scatter diagram of oxygen purity versus hydrocarbon

level from Table 11-1.

90

88

86

0.85 0.95

92

94

96

98

100

Purity (

y)

1.05 1.15 1.25 1.35 1.45 1.55

Hydrocarbon level (x)

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