# Statistics for Environmental Engineers

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0

124

4

Enterococci

20

104

4

Male-specific coliphage

4

75

49

Somatic coliphage

1

78

49

Total culturable viruses

2

122

4

Source: K. M. Doherty (2001). M. S. Project Report, CEE Dept., Tufts University.

19.5 Operations. The maintenance division says that at least 40% of alarms at pump stations are urgent and uses this claim to support a request for increased staffing. Management collected data on the alarms and found that only 49 of 150 were urgent and that staffing was adequate to service these. Can the null hypothesis p > 0.40 be rejected at the 99% level of probability?

# 20

## Multiple Paired Comparisons of k Averages

KEY WORDS data snooping, data dredging, Dunnett’sprocedure, multiple comparisons, sliding reference distribution, studentized range, t-tests, Tukey’s procedure.

The problem of comparing several averages arises in many contexts: compare five bioassay treatments against a control, compare four new polymers for sludge conditioning, or compare eight new combinations of media for treating odorous ventilation air. One multiple paired comparison problem is to compare all possible pairs of k treatments. Another is to compare k — 1 treatments with a control.

Knowing how to do a t-test may tempt us to compare several combinations of treatments using a series of paired t-tests. If there are k treatments, the number of pair-wise comparisons that could be made is k(k — 1)/2. For k = 4, there are 6 possible combinations, for k = 5 there are 10, for k = 10 there are 45, and for k = 15 there are 105. Checking 5, 10, 45, or even 105 combinations is manageable but not recommended. Statisticians call this data snooping (Sokal and Rohlf, 1969) or data dredging (Tukey, 1991). We need to understand why data snooping is dangerous.

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