Population proportion
In statistics a population proportion, generally denoted by
,[1] 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.
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
[2][3] 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.
[5][2] 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.
critical value of the standard normal distribution for a level of confidence
[6] 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.
could fall in between the values of: In general the formula used for estimating a population proportion requires substitutions of known numerical values.
However, these numerical values cannot be "blindly" substituted into the formula because statistical inference requires that the estimation of an unknown parameter be justifiable.
For a more detailed look into regions where this simplification is not used look to (https://en.wikipedia.org/wiki/Binomial_proportion_confidence_interval#Jeffreys_interval ) Suppose a presidential election is taking place in a democracy.
A political scientist wants to determine what percentage of the voter population support candidate B.
To answer the political scientist's question, a one-sample proportion in the Z-interval with a confidence level of 95% can be constructed in order to determine the population proportion of eligible voters in this democracy that support candidate B.
Before a confidence interval is constructed, the conditions for inference will be verified.
With the conditions for inference verified, it is permissible to construct a confidence interval.
By examining a standard normal bell curve, the value for
can be determined by identifying which standard score gives the standard normal curve an upper tail area of 0.0250 or an area of 1 – 0.0250 = 0.9750.
can also be found through a table of standard normal probabilities.
From a table of standard normal probabilities, the value of
can now be substituted into the formula for one-sample proportion in the Z-interval:
Based on the conditions of inference and the formula for the one-sample proportion in the Z-interval, it can be concluded with a 95% confidence level that the percentage of the voter population in this democracy supporting candidate B is between 63.429% and 72.571%.
A commonly asked question in inferential statistics is whether the parameter is included within a confidence interval.
Referring to the example given above, the probability that the population proportion is in the range of the confidence interval is either 1 or 0.
The main purpose of a confidence interval is to better illustrate what the ideal value for a parameter could possibly be.
The level of confidence is based on a measure of certainty, not probability.
