Statistics for Environmental Engineers

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x


У1


У2


Уз


2


0.0


1.7


2.0


5


4.0


2.0


4.5


8


5.1


4.1


5.8


12


8.1


8.9


8.4


15


9.2


8.3


8.8


18


11.3


9.5


10.9


20


11.7


10.7


10.4

39.4 Range of Data. Fit a straight-line calibration model to the first 10 observations in the Exercise


36.3 data set, that is for COD between 60 and 195 mg/L. Then fit the straight line to the full data set (COD from 60 to 675 mg/L). Interpret the change in R for the two cases.

40

Regression Analysis with Categorical Variables


KEY WORDS acid rain, pH, categorical variable, F test, indicator variable, east squares, linear model, regression, dummy variable, qualitative variables, regression sum of squares, t-ratio, weak acidity.


Qualitative variables can be used as explanatory variables in regression models. A typical case would be when several sets of data are similar except that each set was measured by a different chemist (or different instrument or laboratory), or each set comes from a different location, or each set was measured on a different day. The qualitative variables — chemist, location, or day — typically take on discrete values (i.e., chemist Smith or chemist Jones). For convenience, they are usually represented numerically by a combination of zeros and ones to signify an observation’s membership in a category; hence the name categorical variables.


One task in the analysis of such data is to determine whether the same model structure and parameter values hold for each data set. One way to do this would be to fit the proposed model to each individual data set and then try to assess the similarities and differences in the goodness of fit. Another way would be to fit the proposed model to all the data as though they were one data set instead of several, assuming that each data set has the same pattern, and then to look for inadequacies in the fitted model.

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