A population proportion is the proportion (percentage) of a population that has a specified attribute. For example, if the population under consideration consists of all Americans and the specified attribute is “retired, “the population proportion is the proportion of all Americans who are retired. Statisticians often need to determine the proportion (percentage) of a population that has a specified attribute.

Some examples are

•The percentage of U.S. adults who have health insurance

•The percentage of cars in the United States that are imports •The percentage of U.S. adults who favor stricter clean air health standards •The percentage of Canadian women in the labor force.

In the first case, the population consists of all U.S. adults and the specified attribute is “has health insurance. “ For the second case, the population consists of all cars in the United States and the specified attribute is “is an import. “The population in the third case is all U.S. adults and the specified attribute is “favors stricter clean air health standards. “In the fourth case, the population consists of all Canadian women and the specified attribute is “is in the labor force. “We know that it is often impractical or impossible to take a census of a large population. In practice, therefore, we use data from a sample to make inferences about the population proportion. We introduce proportion notation and terminology in the next example. Consider a population in which each member either has or does not have a specified attribute. Then we use the following notation and terminology.

Population proportion: The proportion (percentage) of the entire population that has the specified attribute.

Sample proportion: The proportion (percentage) of a sample from the population that has the specified attribute.

To make inferences about a population mean, μ, we must know the sampling distribution of the sample mean, that is, the distribution of the variable. The same is true for proportions: To make inferences about a population proportion, we need to know the sampling distribution of the sample proportion, that is, the distribution of the variable. Because a proportion can always be regarded as a mean, we can use our knowledge of the sampling distribution of the sample mean to derive the sampling distribution of the sample proportion. The margin of error is equal to half the length of the confidence interval. It represents the precision with which a sample proportion estimates the population proportion at the specified confidence level. In general, the margin of error of an estimator represents the precision with which it estimates the parameter in question. If the margin of error and confidence level is given, then we must determine the sample size required to meet those specifications.

Reference:

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