page_title: Population proportion
section_title: Proof
hierarchical_section_title: Population proportion / Estimation / Proof
caption_attribution_description: English: Bell curve for a sampling distribution of sample proportions
context_page_description: A population proportion, generally denoted by or the Greek letter, is a parameter that describes a percentage value associated with a population. 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 census can be conducted in order to determine the actual value of a population parameter, but often a census is not practical due to its costs and time consumption.
  A population proportion is usually estimated through an unbiased sample statistic obtained from an observational study or experiment. 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 72% of the 2,000 adults sampled did not understand what a gross domestic product is. The value of 72% is a sample proportion. The sample proportion is generally denoted by and in some textbooks by.
context_section_description: In order 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  and its standard deviation is denoted as . Since the value of  is unknown, an unbiased statistic  will be used for . The mean and standard deviation are rewritten as  and  respectively. Invoking the central limit theorem, the sampling distribution of sample proportions is approximately normal—provided that the sample is reasonably large and unskewed. 
  Suppose the following probability is calculated: , where  and  are the standard critical values. 
  The inequality  can be algebraically re-written as follows:
  From the algebraic work done above, it is evident from a level of certainty  that could fall in between the values of .
caption: The sampling distribution of sample proportions is approximately normal when it satisfies the requirements of the Central Limit Theorem.
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