Download Anatomy: Proportion

ANATOMY OF A PROPORTION
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There are instances when we are not interested in a single value (x), or a sample mean, x , but are interested in the Proportion of a
^
population (denoted p), or of a sample (denoted p ; p-hat), that possesses a specific characteristic of interest to us.
An example for a population:
Suppose there are 789,654 Peanut Butter M&M’s in a large container waiting to
be bagged, of which 157,931 are yellow.
Then the proportion of all yellow Peanut Butter M&M’s in this container is
p
157931
 .20
789654
The population proportion p is equal to the number of elements in the population
with a specific characteristic (here, yellow in color), divided by the total number
of elements in the population.
An example for a sample:
Now suppose a random sample of 240 Peanut Butter M&M’s is taken from this
container and 52 of them are yellow.
Then the sample proportion of yellow Peanut Butter M&M’s is
^
p
52
 .22
240
The sample proportion p is equal to the number of elements in the sample with a
specific characteristic (here, yellow in color), divided by the total number of
elements in the sample.
Why the difference
(.20 versus .22)?
Recall that there is no guarantee that a randomly
drawn sample will exactly match the population
parameter. That difference is due to sampling error,
a natural part of the sampling process. Recall also
that with a larger sample size our sample proportion
would be closer to the true population proportion p.