In statistics a population proportion, generally denoted by P {\displaystyle P} or the Greek letter π {\displaystyle \pi } , is a parameter that describes a percentage value associated with a population. A census can be conducted to determine the actual value of a population parameter, but often a census is not practical due to its costs and time consumption. For example, the 2010 United States census showed that 83.7% of the American population was identified as not being Hispanic or Latino; the value of .837 is a population proportion. In general, the population proportion and other population parameters are unknown. A population proportion is usually estimated through an unbiased sample statistic obtained from an observational study or experiment, resulting in a sample proportion, generally denoted by p ^ {\displaystyle {\hat {p}}} and in some textbooks by p {\displaystyle p} . For example, the National Technological Literacy Conference conducted a national survey of 2,000 adults to determine the percentage of adults who are economically illiterate; the study showed that 1,440 out of the 2,000 adults sampled did not understand what a gross domestic product is. The value of 72% (or 1440/2000) is a sample proportion.
Mathematical definition
A proportion is mathematically defined as being the ratio of the quantity of elements (a countable quantity) in a subset S {\displaystyle S} to the size of a set R {\displaystyle R} :
P = X N , {\displaystyle P={\frac {X}{N}},}
where X {\displaystyle X} is the count of successes in the population, and N {\displaystyle N} is the size of the population. This mathematical definition can be generalized to provide the definition for the sample proportion:
p ^ = x n {\displaystyle {\hat {p}}={\frac {x}{n}}}
where x {\displaystyle x} is the count of successes in the sample, and n {\displaystyle n} is the size of the sample obtained from the population.
Estimation
One of the main focuses of study in inferential statistics is determining the "true" value of a parameter. Generally the actual value for a parameter will never be found, unless a census is conducted on the population of study. However, there are statistical methods that can be used to get a reasonable estimation for a parameter. These methods include confidence intervals and hypothesis testing. Estimating the value of a population proportion can be of great implication in the areas of agriculture, business, economics, education, engineering, environmental studies, medicine, law, political science, psychology, and sociology. A population proportion can be estimated through the usage of a confidence interval known as a one-sample proportion in the Z-interval whose formula is given below:
p ^ ± z ∗ p ^ ( 1 − p ^ ) n {\displaystyle {\hat {p}}\pm z^{*}{\sqrt {\frac {{\hat {p}}(1-{\hat {p}})}{n}}}}
where p ^ {\displaystyle {\hat {p}}} is the sample proportion, n {\displaystyle n} is the sample size, and z ∗ {\displaystyle z^{*}} is the upper 1 − C 2 {\displaystyle {\frac {1-C}{2}}} critical value of the standard normal distribution for a level of confidence C {\displaystyle C} .
Proof To derive the formula for the one-sample proportion in the Z-interval, a sampling distribution of sample proportions needs to be taken into consideration. The mean of the sampling distribution of sample proportions is usually denoted as μ p ^ = P {\displaystyle \mu _{\hat {p}}=P} and its standard deviation is denoted as:
σ p ^ = P ( 1 − P ) n {\displaystyle \sigma _{\hat {p}}={\sqrt {\frac {P(1-P)}{n}}}}
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