Statistics for Environmental Engineers

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These matrices are easy to create and manipulate for a factorial experimental design. X is an orthogonal matrix, that is, the inner product of any two columns of vectors is zero. Because X is an orthogonal matrix, X’X is a diagonal matrix, that is, all elements are zero except diagonal elements. If X has n columns and m rows, X’ has m columns and n rows. The product X’X will be a square matrix with n rows and n columns. If X’X is a diagonal matrix, its inverse (X’X)-1 is just the reciprocal of the elements of X’ X.

Case Study Solution


The variability of the nitrate measurements is larger at the higher concentrations. This is because the logarithmic scale of the instrument makes it possible to read to 0.1 mg/L at the low concentration but only to 1 mg/L at the high level. The result is that the measurement errors are proportional to the measured concentrations. The appropriate transformation to stabilize the variance in this case is to use the natural logarithm of the measured values. Each value was transformed by taking its natural logarithm and then the logs of the replicates were averaged.

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