Math160.Section8.2.pdf - Math A160 - Cooley Introduction to...

Math A160 - Cooley Introduction to Statistics OCC
SECTION 8.2 – Distribution of the Sample Proportion
- 1 -
Definition
Suppose that a random sample of size n is obtained from a population in which each individual either does or
does not have a certain characteristic. The sample proportion, denoted ˆ p (read “p-hat”), is given by
ˆ
p =
x
n
where x is the number of individuals in the sample with the specified characteristic. The sample proportion, ˆ p ,
is a statistic that estimates the population proportion, p .
Sampling Distribution of
ˆ
p
For a simple random sample of size n with a population proportion p ,
• The mean of the sampling distribution of
ˆ
p is µ
ˆ p
= p .
• The standard deviation of the sampling distribution of
ˆ
p is
ˆ
(1 )
p
p p
n
σ
−
=
.
• The shape of the sampling distribution of
ˆ
p is approximately normal provided np (1 – p ) ≥ 10.
Note: We also require that the sampled values must be independent of each other; that is, one outcome cannot
affect the success or failure of any other outcome. So, when sampling from finite populations, this
means that the sample size can be no more than 5% of the population size ( n ≤ 0.05 N ).