# Statistics for Environmental Engineers

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This is the value given in Table 19.3 for x = 1 and p = 0.05. Likewise, the probability of two or less deaths is Pr(x < 2) = 0.92 and the probability of exactly two deaths is:

Pr(x = 2) = Pr(x < 2) — Pr(x < 1) = 0.92 — 0.74 = 0.18

Table 19.3 shows that the range of possible outcomes in a binomial process can be quite large and that the variability about the expected value increases as p increases. This characteristic of the variance needs to be considered in designing bioassay experiments.

Most binomial tables only provide for up to n = 20. Fortunately, under certain circumstances, the normal distribution provides a good approximation to the binomial distribution. The binomial probability distribution is symmetric when p = 0.5. The distribution approaches symmetry as n becomes large, the approach being more rapid when p is close to 0.5. For values of p < 0.5, it is skewed to the right; for p > 0.5, it is skewed left. For large n, however, the skewness is not great unless p is near to 0 or 1.

As n increases, the binomial distribution can be approximated by a normal distribution with the same mean and variance [nx = np and a2x = np(1 — p)]. The normal approximation is reasonable if np > 5 and n(1 — p) > 5. Table 19.3 and Figure 19.1 show that the distribution is exactly symmetric for n = 20 and p = 0.5. For n = 20 and p = 0.2, where np(1 — p) is only 3.2, the distribution is approaching symmetry.

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