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Sample Proportion Estimator
The proportion of individuals with a characteristic in a population, p , is estimated by a
sample proportion estimator, which simply calculates the proportion of that characteristic
in the sample:
C
n
where C is the number of subjects in the sample which is observed to have the
charactestic in question.
p= Utku Suleymanoglu (UMich) Sampling Distributions 17 / 21 Sampling Distribution of P ¯
As X , P changes sample to sample, so it is a random variable.
Sampling Distribution of P
P has a sampling mean of:
E (P ) = p
and sampling variance:
V (P ) = p (1 − p )
n where p is the population proportion.
Furthermore, CLT is helpful for sample proportions: As long as np (1 − p ) ≥ 5, you can
approximate the distribution of P with a normal distribution with mean p and variance
p (1 − p )/n.
In class, we talked about how the population values cannot be considered normally
distributed in this case, and that they are in fact distributed with a Bernoulli distribution.
¯
This makes C a Binomial(n,p) random variable. Using this we showed how E (P ) and
¯
V (P ) formulas above were derived.
¯
(Why does CLT work here: is P a sam...
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 Spring '08
 STAFF

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