Advanced High-School Mathematics

392 CHAPTER 6 Inferential Statistics

The reason is the highly skewed nature of a binomial population with
parameterpvery close to either 0 or 1, meaning that the Central Limit
Theorem will need much larger samples before the distribution starts
to become acceptably normal. A proposed modification^27 is to replace

pˆin the above interval by the new statisticp∗=
x+ 2
n+ 4

, wherexis the

measured number of type A members in the sample andnis the sample

size. Also, the sample standard deviation

Ã

pˆ(1−pˆ)

n

is replaced by

the expression

Ã

p∗(1−p∗)

n+ 4

. The resulting confidence interval performs

better for the the full range of possibilities forp, even whennis small!

Rule of Thumb: Since the methods of this section rely on the test

statistics
P̂−p
√̂
P(1−P̂)/n

being approximately normally distributed, any

sort of guidance which will help us assess this assumption will be help-
ful. One typically used one is that if the approximate assumption of
normality is satisfied, then ˆp±three sample standard deviations should
both lie in the interval (0,1). Failure of this to happen indicates that
the sample size is not yet large enough to counteract the skewness in the
binomial distribution. That is to say, we may assume that the methods
of this section are valid provided that

0 <pˆ± 3

Ã

pˆ(1−pˆ)

n

< 1.

6.4.4 Sample size and margin of error

In the above discussions we have seen the our confidence intervals had
the form

Estimate ± Margin of Error

at a given confidence level. We have also seen that decreasing the

(^27) See Agresti, A., and Coull, B.A.,Approximate is better than ‘exact’ for interval estimation of
binomial proportions,The American Statistician, Vol. 52, N. 2, May 1998, pp. 119–126.