Wiki Page, Math 13 - Perez Sandbox : Module 18 - Introduction to Statistical Inference (1b of 3)

Wiki Page
2026-01-17T21:15:23-08:00

Summary

A sample proportion from a random sample provides a reasonable estimate of the population proportion. We do not expect the sample proportion to be exactly equal to the population proportion, but we expect the sample population proportion to be somewhat close to the population proportion. The purpose of confidence intervals is to use the sample proportion to construct an interval of values that we can be reasonably confident contains the true population proportion.

What Is the Connection to the Sampling Distribution?

Sample proportions are estimates for the population proportion, so each sample proportion has error. For an individual sample, we will not know the exact amount of error, so we report a margin of error based on the standard error. Recall that the standard error is the standard deviation of sampling distribution. We can view the standard error as the typical or average error in the sample proportions. To see how this works, let’s return to a familiar sampling distribution.

Recall our previous investigation of gender in the population of part-time college students. We investigated these questions: What proportion of part-time college students are female? If we predict that the proportion is 0.60, how much error can we expect to be confident of in our prediction?

We predicted the population proportion was 0.60 and ran a simulation to examine the variability in sample proportions for samples of 100 part-time college students. Here is the sampling distribution from the simulation.

We see that we can be very confident that most samples of this size will have proportions that differ from 0.60 by at most 2 standard errors. For this simulation, the standard error in sample proportions was about 0.049. About 95% of the samples have an error less than 2(0.049) = 0.098.

If we use two standard errors as the margin of error, we can rewrite the confidence interval.

sample statistic margin of error

sample proportion 2(standard errors)

sample proportion 2(0.049)

sample proportion 0.098

Different sample proportions give different intervals, but all have the same margin of error. Here are some examples:

If the sample proportion is 0.57, the confidence interval is 0.472 to 0.668. Here are our calculations.

sample proportion ± margin of error

0.57 2(0.049)

0.57 0.098

The endpoints of the interval are 0.57 – 0.098 = 0.472 and 0.57 + 0.098 = 0.668. The confidence interval is 0.472 to 0.668.

If the sample proportion is 0.66, the confidence interval is 0.562 to 0.758. Here are our calculations.

sample proportion ± margin of error

0.66 2(0.049)

0.66 0.098

The endpoints of the interval are 0.66 – 0.098 = 0.562 and 0.66 + 0.098 = 0.758. The confidence interval is 0.562 to 0.758.

We say that we are 95% confident that each interval contains the population proportion.

We are 95% confident that between 47.2% and 66.8% of part-time college students are female.

We are 95% confident that between 56.2% and 75.8% of part-time college students are female.

Obviously, these two confidence intervals give different estimates for the population proportion. For example, the first interval suggests that it is possible that 52% of all part-time college students are female because 52% lies within that interval. But the second interval suggests that 52% is a NOT a good estimate for the population proportion because this interval does not contain 52%.

In statistical inference, we are not 100% confident that the interval gives us accurate bounds on the population proportion. But we can say that we are 95% confident that it does. In other words, 95% of the time the population proportion lies within the interval. This why we say we are 95% confident that the population proportion lies within the interval and why we call these intervals confidence intervals.

Mark this page as done, then click the "Next" or > button to continue.

Content by the Open Learning Initiative and licensed under CC BY.