Wiki Page, Math 13 - Perez Sandbox : Module 19 - Estimating a Population Proportion (3 of 3)

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2026-01-17T21:15:26-08:00

Confidence Interval Width

The goal of the confidence interval is to estimate the population proportion. If the interval contains the population proportion, a small amount of error means we have a more precise estimate. Narrower confidence intervals give more precise interval estimates for the population proportion, but this is true only if the intervals contain the population proportion.

For a given standard error, a lower confidence level corresponds to a smaller margin of error. In general, 90% confidence intervals are narrower than 95% confidence intervals because there is a smaller margin of error. But we are less confident that 90% confidence intervals contain the population proportion. Recall that in the long run, 10% of these intervals will not contain the population proportion at all! We therefore have to choose between precision and confidence.

Of course, ideally, we would like to have a narrow interval and a high level of confidence. We can achieve this by increasing the size of the sample.

EXAMPLE College students and marijuana legalization

National surveys show that about 43% of American adults support the legalization of marijuana. What proportion of students at Capital Community College support the legalization of marijuana? Suppose students conduct two surveys. For one survey, they randomly select a sample of 100 students. For the other survey, they randomly select a sample of 400 students. Surprisingly, in both surveys, the proportion in favor of legalization is 55%. The students calculate the 95% confidence interval for both surveys.

What is the impact of the size of the sample on the confidence interval?

For the sample of size 100, the confidence interval is

For the sample of size 400, the confidence interval is

The larger sample size gives a smaller margin of error.

This makes sense. Our intuition tells us that larger samples should give more better estimates of the population proportion.

We also saw know that sample size affects the variability in the sample proportions. Proportions from larger samples vary less. If there is less variability in the sampling distribution, the standard error is smaller. Since we use the standard error to find the margin of error, larger samples will produce a smaller margin of error.

In general, to decrease the margin of error, we can increase the sample size or decrease the confidence level. We always prefer to increase the sample size because it allows us to keep a higher level of confidence. We want a higher level of confidence because the confidence level is the proportion of intervals that actually contains the population proportion.

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