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Probabiltiy of Observing Exactly x Deaths in a Group of 20 Trials
Probability of death
of a single organism
0.2
p=0.05
0
0.2
p=0.10
0

0.2
p=0.15
0

0.2
p=0.20
0

0.2
p=0.25
0

0.2
p=0.5
0
0 5 10 15 20
x

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 (^) is used to distinguish the population parameter p and the sample
proportion, which will be called p ˆ = x/n. The hat (^) 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:

s x/n = p 1 – p)/n
(
2
Two independent test groups of size n 1 and n 2 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 x 1 in the control and x 2 in the treatment, giving observed sample proportions of
= x 2 /n 2 . If we assume, as in a null hypothesis, that the control and treatment populations
p ˆ 1 = x 1 /n 1 and p ˆ 2
will be
have the same true underlying population probability p (i.e., p = p 1 = p 2 ), then p ˆ 1 and p ˆ 2
normally distributed with mean p and variances p(1 − p)/n 1 and p(1 − p)/n 2 , respectively.
− p ˆ 2 ) will be normally distributed with mean
The difference between the two sample proportions (p ˆ 1
zero and variance:
(
(
p)
1
1
p 1 –

--------------------- + p 1 – p) p 1 – p) ----- + -----  
(
--------------------- =

n 1 n 2 n 1 n 2
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