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l1592_frame_Ch23 Page 206 Tuesday, December 18, 2001 2:44 PM

with curves for β = 0.05, 0.1, and 0.2. The abscissa is the standardized range, τ = (µ max − µ min )/σ, where
µ max and µ min are the planning values for the largest and smallest mean values, and σ is the planning
standard deviation.

Example 23.6

How small a difference can be detected between five groups of contaminated soil with a sample
size of n = 10, assuming for planning purposes that σ = 2.0, and for risk levels α = 0.05 and β =
0.1? Read from the graph (τ = 1.85) and calculate:
µ max µ min = τσ = 1.85 2.0) = 3.7
(
–

Sample Size to Estimate a Binomial Population
A binomial population consists of binary individuals. The horses are black and white, the pump is faulty
or fault-free, an organism is sick or healthy, the air stinks or it does not. The problem is to determine
how many individuals to examine in order to estimate the true proportion p for each binary category.
An approximate expression for sample size of a binomial population is:
∗ 
z α/2
n = p 1 – p ) ------- 2
∗
(
 E 
∗
where p is the a priori estimate of the proportion (i.e., the planning value). If no information is available
∗
from prior data, we can use p = 1/2, which will give the largest possible n, which is:

n = 1 z α/2 2
--- --------
4 E 

This sample size will give a (1 − α)100% confidence interval for p with half-length E. This is based on
a normal approximation and is generally satisfactory if both np and n(1 − p) exceed 10 (Johnson, 2000).
Example 23.7

Some unknown proportion p of a large number of installed devices (i.e., flow meters or UV lamp
ballasts) were assembled incorrectly and have to be repaired. To assess the magnitude of the
problem, the manufacturer wishes to estimate the proportion ( p) of installed faulty devices. How
many units must be examined at random so that the estimate p ˆ () will be within ±0.08 of the
true proportion p, with 95% confidence? Based on consumer complaints, the proportion of faulty
devices is thought to be less than 20%.
∗
In this example, the planning value is p = 0.2. Also, E = 0.08, α = 0.05, and z 0.025 = 1.96, giving:

1.96
(
n = 0.2 1 0.2) ----------  2 ≈ 96
–
 0.08
If fewer than 96 units have been installed, the manufacturer will have to check all of them.
(A sample of an entire population is called a census.)
The test on proportions can be developed to consider type I and type II errors. There is typically large
inherent variability in biological tests, so bioassays are designed to protect against the two kinds of
decision errors. This will be illustrated in the context of bioassay testing where raw data are usually
converted into proportions.
The proportion of organisms affected is compared with the proportion of affected organisms in an
unexposed control group. For simplicity of discussion, assume that the response of interest is survival
© 2002 By CRC Press LLC