Statistics for Environmental Engineers

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Probability of death of a single organism




I 1 1 1 1 I 1 1 1 1 I





FIGURE 19.1 The binomial distribution with n = 20 for several values of p.

Assessing the Difference Between Two Proportions

The binomial distribution expresses the number of occurrences of an event x in n trials, where p is the probability of occurrence in a single trial. Usually the population probability p in a binomial process is unknown, so it is often more useful to examine the proportion of occurrences rather than their absolute number, x. Contrary to our guidelines on notation (Chapter 2), the population parameter p is not denoted with a Greek letter symbol. A hat (л) is used to distinguish the population parameter p and the sample proportion, which will be called p = x/n. The hat (л) is a reminder that the sample proportion is a statistic computed from the data and that it estimates the population proportion. The sample proportion (x/n) is an unbiased estimator of the underlying population probability ( p) in a binomial process.

The sample variance of p is:

sX/n = p( 1p )/n

Two independent test groups of size n1 and n2 are to be compared. Suppose that group 1 represents a control (no exposure) and group 2 is the treatment group (i.e., exposed to effluent). The number of surviving organisms is x1 in the control and x2 in the treatment, giving observed sample proportions of p1 = x1/n1 and p2 = x2/n2. If we assume, as in a null hypothesis, that the control and treatment populations have the same true underlying population probability p (i.e., p = p1 = p2), then p1 and p2 will be normally distributed with mean p and variances p(1 — p)/n1 and p(1 — p)/n2, respectively.

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