Wiki Page, Math 13 - Perez Sandbox : Module 19 - Wrap Up "Estimating a Population Proportion"-2

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2026-01-17T21:15:27-08:00

Let’s Summarize

The sample proportion is a point estimate for the population proportion, but sample proportions vary. We therefore use an interval estimate, called a confidence interval, to give us a range of values for the population proportion.

We can calculate a confidence interval for a population proportion when we can use a normal distribution to model the long-run behavior of sample proportions. We can use a normal distribution model when the sample is randomly selected and there are at least 10 observed successes and 10 observed failures in the sample.

The interpretation of a confidence interval depends on the confidence level. For example, using a 95% confidence level, we are 95% confident that the population proportion falls within the interval.

95% confident means that in the long run 95% of random samples will generate intervals that contain the population proportion and 5% will not.

A confidence interval is a sample proportion plus or minus a margin of error. The margin of error is related to the confidence level. For a 95% confidence level, the margin of error is approximately two standard errors. The formula is

where is 1.645 for 90% confidence, 1.96 for 95% confidence and 2.58 for 99% confidence.

Lower confidence levels and higher sample sizes lead to narrower confidence intervals. A narrower confidence interval has a smaller error. Since we want to be confident that an interval accurately estimates the population proportion, high levels of confidence are desirable. So larger sample sizes are the preferred way to decrease the error and create narrower confidence intervals.

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