Wiki Page, Math 13 - Perez Sandbox : Module 18 - Introduction to Statistical Inference (2a of 3)

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2026-01-17T21:15:24-08:00

A Look at 95% Confidence Intervals on the Number Line

Let’s look again at the formula for a 95% confidence interval.

The lower end of the confidence interval is sample proportion – 2(standard error).

The upper end of the confidence interval is sample proportion + 2(standard error).

Every confidence interval defines an interval on the number line that is centered at the sample proportion. For example, suppose a sample of 100 part-time college students is 64% female. Here is the 95% confidence interval built around this sample proportion of 0.64.

We know the margin of error in a confidence interval comes from the standard error in the sampling distribution. For a 95% confidence interval, the margin of error is equal to 2 standard errors. This is shown in the following diagram.

The width of the interval is the same as the width of the middle 95% of the sampling distribution. The next diagram illustrates this relationship.

When Does a 95% Confidence Interval Contain the True Population Proportion?

If the sample proportion has an error that is less than 2 standard errors, then the 95% confidence interval built around this sample proportion will contain the population proportion.

The sample proportion 0.64 is within 2 standard errors of 0.60, so 0.60 is in the 95% confidence interval built around 0.64.

In the following figure, the sample proportion 0.72 is not within 2 standard errors of 0.60, so 0.60 is not in the 95% confidence interval built around 0.72.

How Confident Are We That a 95% Confidence Interval Contains the Population Proportion?

Following are three confidence intervals for estimating the proportion of part-time college students who are female. We are confident that most of these intervals will contain the population proportion, like the green intervals shown here. But some will not contain the population proportion, like the red interval shown here.

Of course, we don’t know the population proportion (which is why we want to estimate it with a confidence interval!). In reality, we cannot determine if a specific confidence interval does or does not contain the population proportion; that’s why we state a level of confidence. For these intervals, we are 95% confident that an interval contains the population proportion. In other words, 95% of random samples of this size will give confidence intervals that contain the population proportion. The sad news is that we never know if a particular interval does or does not contain the unknown population proportion.

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