A population proportion, generally denoted by
Contents
- Mathematical definition
- Estimation
- Proof
- Conditions for inference
- Example
- Solution
- Value of the parameter in the confidence interval range
- Common errors and misinterpretations from estimation
- References
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
Mathematical definition
A proportion is mathematically defined as being the ratio of the values in a subset
As such, the population proportion can be defined as follows:
This mathematical definition can be generalized to provide the definition for the sample proportion:
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:
Proof
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
Suppose the following probability is calculated:
The inequality
Conditions for inference
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. In order for a parameter's estimation to be justifiable, there are three conditions that need to be verified:
- The data's individual observation have to be obtained from a simple random sample of the population of interest.
- The data's individual observations have to display normality. This can be verified mathematically with the following definition:
- Let
n be the sample size of a given random sample and letp ^ n p ^ ≥ 10 andn ( 1 − p ^ ) ≥ 10 , then the data's individual observations display normality. - The data's individual observations have to be independent of each other. This can be verified mathematically with the following definition:
- Let
N be the size of the population of interest and letn be the sample size of a simple random sample of the population. IfN ≥ 10 n , then the data's individual observations are independent of each other.
The conditions for SRS, normality, and independence are sometimes referred to as the conditions for the inference tool box in most statistical textbooks.
Example
Suppose a presidential election is taking place in a democracy. A random sample of 400 eligible voters in the democracy's voter population shows that 272 voters support candidate B. 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.
Solution
It is known from the random sample that
Before a confidence interval is constructed, the conditions for inference will be verified.
The condition for normality has been met.
The population size
Hence, the condition for independence has been met.
With the conditions for inference verified, it is permissible to construct a confidence interval.
Let
To solve for
By examining a standard normal bell curve, the value for
From a table of standard normal probabilities, the value of
The values for
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 that support candidate B is between 63.429% and 72.571%.
Value of the parameter in the confidence interval range
A commonly asked question in inferential statistics is whether the parameter is included within a confidence interval. The only way to answer this question is for a census to be conducted. 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. That is, the parameter is included in the interval range or it is not. The main purpose of a confidence interval is to better illustrate what the ideal value for a parameter could possibly be.
Common errors and misinterpretations from estimation
A very common error that arises from the construction of a confidence interval is the belief that the level of confidence such as