# Statistics for Environmental Engineers

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6.77

6.87

6.83

6.84

6.77

6.76

6.73

6.80

7.01

6.67

6.85

6.90

6.95

6.88

6.73

6.92

6.76

6.68

6.79

6.93

6.86

6.87

6.95

6.73

6.59

6.84

6.62

6.77

6.53

6.94

6.91

6.90

6.75

6.74

6.74

6.76

6.65

6.72

6.87

6.92

6.98

6.70

6.97

6.95

6.94

6.93

6.80

6.84

6.78

6.67

Note: Time runs from left to right.

FIGURE 6.1 Time series plot of the pH data with the moving average of ten consecutive values.

Observation

FIGURE 6.2 Ten-day moving averages of pH.

10-day Moving Average of pH

FIGURE 6.3 External reference distribution for ten-day moving averages of pH.

FIGURE 6.4

-0.12 -0.08 -0.04    0    0.04 0.08

Difference of 10-day Moving Averages10 Days Apart

External reference distribution for differences of 10-day moving averages of pH.

### Using a Reference Distribution to Compare Two Mean Values

Let the situation in the previous example change to the following. An experiment to evaluate the effect of an industrial discharge into a treatment process consists of making 10 observations consecutively before any addition and 10 observations afterward. We assume that the experiment is not affected by any transients between the two operating conditions. The average of 10 consecutive pre-discharge samples was 6.80, and the average of the 10 consecutive post-discharge samples was 6.86. Does the difference of 6.80 — 6.86 = -0.06 represent a significant shift in performance?

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