AP Stats Proportion Confidence Interval Calculator

Estimate a population proportion from a sample. Enter the number of successes and the sample size, choose a confidence level, and this AP Stats Proportion Confidence Interval Calculator gives p-hat, the interval, and the margin of error.

What Is a Confidence Interval for a Proportion?

A confidence interval for a proportion gives a range of plausible values for the true proportion in a population, based on a sample proportion. For example, if 60% of 400 surveyed students support a change, a 95% interval might run from about 55% to 65%. The interval is only valid when the sample is random and both the number of successes and the number of failures are at least 10.

How to Use This Calculator

1. Count successes x in your sample, for example the number of students who passed. 2. Enter the total sample size n. 3. Choose 90%, 95%, or 99% confidence. 4. Read p-hat, the interval, and the margin of error, then write your conclusion in context: "We are 95% confident that the true proportion is between..."

How to Calculate a Proportion Confidence Interval

Formulas: p̂ = x / n, standard error = √[p̂(1 − p̂) / n], and the interval is p̂ ± z* × standard error, where z* is 1.645 for 90%, 1.96 for 95%, and 2.576 for 99% confidence.

Worked example: x = 120, n = 200, 95% confidence. p̂ = 120 / 200 = 0.60. Standard error = √[0.60 × 0.40 / 200] ≈ 0.0346. Margin of error = 1.96 × 0.0346 ≈ 0.0679. Interval = 0.60 ± 0.0679 = (0.532, 0.668). Conditions: np̂ = 120 and n(1 − p̂) = 80, both at least 10, so the normal approximation is reasonable. Try these values in the calculator.

Tips and Common Mistakes

Do not use the population proportion p in the standard error when it is unknown; use p̂ from the sample. Always check the large-counts conditions, np̂ ≥ 10 and n(1 − p̂) ≥ 10, and random sampling. A 95% confidence interval does not mean there is a 95% probability the true proportion is in this one interval; it means 95% of intervals made this way would capture the true proportion.

Frequently Asked Questions

What is p-hat?

p-hat is the sample proportion, x divided by n. It is the point estimate for the unknown population proportion p.

Why does a higher confidence level give a wider interval?

A higher confidence level uses a larger z*, so the margin of error grows. You gain confidence but lose precision.

What if np-hat or n(1 - p-hat) is less than 10?

The normal approximation may not be reliable. State that the condition is not met and do not trust the interval as much; a larger sample is usually needed.

What is the difference between a one-proportion z-interval and a two-proportion z-interval?

A one-proportion interval estimates one population proportion. A two-proportion interval estimates the difference between two population proportions and uses a different standard error.

How do I write the conclusion?

Write it in context with the confidence level and the interval, for example: "We are 95% confident that the true proportion of students who passed is between 0.532 and 0.668."

What is the TI-84 equivalent?

On a TI-84, use STAT, TESTS, 1-PropZInt, enter x and n, and choose the confidence level. It should match this calculator.

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