*19*

In statistics, a **population proportion** refers to the fraction of individuals in a population with a certain characteristic.

For example, suppose 43.8% of individuals in a certain city support a new law. The valueÂ **0.438** represents a population proportion.

**Formula for a Population Proportion**

A population proportion always ranges between 0 and 1 (or 0% to 100% in percentage terms) and it is calculated as follows:

**p = X / N**

where:

**p:**The population proportion**X:**The count of individuals in a population with a certain characteristic.**N:**The total number of individuals in a population.

**How to Estimate a Population Proportion**

Since it is usually too time-consuming and costly to collect data for every individual in a population, we often collect data for a sample instead.

For example, suppose we want to know what proportion of residents in a certain city support a new law. If the population consists of 50,000 total residents, we may take a simple random sample of 1,000 residents:

We would then calculate the sample proportion as follows:

**pÌ‚ = x / n**

where:

**pÌ‚:**The sample proportion**x:**The count of individuals in the sample with a certain characteristic.**n:**The total number of individuals in the sample.

We would then use this sample proportion toÂ *estimate* the population proportion. For example, if 367 of the 1,000 residents in the sample supported the new law, the sample proportion would be calculated as 367 / 1,000 = **0.367**.

Thus, our best estimate for the proportion of residents in the population who supported the law would beÂ **0.367**.

**Confidence Interval for a Population Proportion**

Although the sample proportion provides us with an estimate of the true population proportion, thereâ€™s no guarantee that the sample proportion will exactly match the population proportion.

For this reason, we typically construct a confidence interval â€“ a range of values that are likely to contain the true population proportion with a high degree of confidence.

The formula to calculate a confidence interval for a population proportion is:

**Confidence Interval = pÌ‚****Â +/-Â z*âˆšpÌ‚(1-pÌ‚) / n**

where:

**pÌ‚:**sample proportion**z:Â**the chosen z-value**n:Â**sample size

The z-value that you will use is dependent on the confidence level that you choose. The following table shows the z-value that corresponds to popular confidence level choices:

Confidence Level | z-value |
---|---|

0.90 | 1.645 |

0.95 | 1.96 |

0.99 | 2.58 |

Notice that higher confidence levels correspond to larger z-values, which leads to wider confidence intervals. This means that, for example, a 95% confidence interval will be wider than a 90% confidence interval for the same set of data.

**Example: Confidence Interval for a Population Proportion**

Suppose we want to estimate the proportion of residents in a cityÂ that are in favor of a certain law. We select a random sample of 100 residents and ask them about their stance on the law. Here are the results:

- Sample sizeÂ
**n = 100** - Proportion in favor of law
**pÌ‚****Â = 0.56**

Here is how to find various confidence intervals for the population proportion:

**90% Confidence Interval:Â **0.56Â +/-Â 1.645*(âˆš.56(1-.56) / 100)Â =Â **[0.478, 0.642]**

**95% Confidence Interval:Â **0.56Â +/-Â 1.96*(âˆš.56(1-.56) / 100)Â =Â **[0.463, 0.657]**

**99% Confidence Interval:Â **0.56Â +/-Â 2.58*(âˆš.56(1-.56) / 100)Â =Â **[0.432, 0.688]**

**Note:Â **You can also find these confidence intervals by using the Confidence Interval for Proportion Calculator.