In this section, you will learn how margin of error controls confidence interval width, shaping how precise your estimates of population proportions can realistically be.

Margin of Error and Interval Width

In one-sample confidence intervals for a population proportion, the margin of error and the width of the confidence interval are tightly connected. For AP Statistics, you should understand not only how to compute them, but also how their relationship affects the precision of your estimate of the population proportion.

In a confidence interval for a population proportion, the estimate is written in the form “point estimate ± margin of error.” The point estimate is the sample proportion, and the margin of error tells you how far to go above and below that estimate to capture plausible values of the true population proportion.

How Margin of Error Determines Interval Width

A confidence interval for a population proportion has two endpoints: the lower bound and the upper bound. The interval width is the distance from one endpoint to the other. Because the structure of the interval is “estimate ± ME,” the distance from the center to either endpoint is exactly the margin of error.

EQUATION

Because the interval extends ME units above and ME units below the center, the total width is twice the margin of error.

This diagram illustrates margin of error as the distance from the center point estimate to one bound, and the full confidence-interval width as the total span across the interval. It reinforces the relationship that interval width equals twice the margin of error. Minor labels related to 95% coverage extend slightly beyond the syllabus focus but remain appropriate for AP Statistics contexts. Source.

This is exactly the relationship stated in the syllabus: width is directly related to margin of error and equals 2 × ME.

Key Relationships to Remember

When thinking about margin of error and interval width, focus on these ideas:

• Direct relationship

  • If ME increases, the interval width increases.

  • If ME decreases, the interval width decreases.

• Width is exactly twice ME

  • The interval runs ME below the sample proportion and ME above it.

  • This means the distance from the lower bound to the upper bound is 2 × ME.

• Precision of the estimate

  • A smaller margin of error gives a narrower interval, which means a more precise estimate of the population proportion.

  • A larger margin of error gives a wider interval, reflecting less precision and more uncertainty.

Interpreting Precision in Context

The syllabus emphasizes how this relationship affects the precision of the interval estimate. Precision is about how tightly the interval pins down the unknown population proportion.

• A narrow interval (small ME) suggests that the sample provides a sharp, focused estimate of the population proportion.

• A wide interval (large ME) suggests that the sample provides a rough, less specific estimate.

When interpreting a confidence interval for a proportion, you should connect your language to this idea of precision:

• If the interval is narrow, you can say that the sample provides a more precise estimate of the true population proportion.

• If the interval is wide, you should acknowledge that there is greater uncertainty about the exact value of the population proportion.

Conceptual Causes of Changes in Interval Width

Even though the detailed mechanics of how different factors affect ME are handled in other subsubtopics, you should conceptually link margin of error and width to typical features of a study:

• Sample size (n)

  • Larger samples usually lead to smaller margin of error and thus narrower intervals, increasing precision.

  • Smaller samples usually lead to larger margin of error and wider intervals, decreasing precision.

• Confidence level

  • Higher confidence levels require a larger critical value z*, which increases ME and interval width.

  • Lower confidence levels use a smaller critical value z*, which decreases ME and interval width.

A wider interval means a larger margin of error and less precision, while a narrower interval means a smaller margin of error and a more precise estimate of the population proportion.

This figure shows how higher confidence levels correspond to wider confidence intervals. Because margin of error is the distance from the center to a cutoff, increasing confidence results in larger margins of error and reduced precision. The labeled critical values extend slightly beyond the exact subsubtopic but align with standard AP Statistics presentation. Source.

Although these ideas are explored in more depth elsewhere, you should see that any factor that increases the margin of error will automatically widen the interval, while any factor that decreases the margin of error will automatically narrow it.

Using Margin of Error and Interval Width to Judge Study Design

Because interval width is directly tied to margin of error, it becomes a practical tool for evaluating or comparing studies:

• When comparing two confidence intervals for population proportions:

  • The study with the smaller margin of error (and therefore narrower interval) provides more precise information about its population.

  • The study with the larger margin of error (and therefore wider interval) provides less precise information.

• When designing a study:

  • Researchers often decide on an acceptable margin of error ahead of time because it directly controls the width of the resulting interval.

  • Choosing a smaller target margin of error means planning for a narrower interval and thus more precision, usually requiring a larger sample size.

Through all of this, the central AP Statistics idea for this subsubtopic is that margin of error and interval width are inseparable: the confidence interval’s width is exactly twice the margin of error, and this simple numerical relationship has important consequences for how precisely we can estimate a population proportion.