6.2 Constructing a Confidence Interval for a Population Proportion

A confidence interval for a population proportion gives you a range of likely values for the true proportion, built as p̂ ± z*√(p̂(1−p̂)/n). To construct one, you use a one-sample z-interval for a proportion, check that your data are independent and that the sampling distribution is approximately normal, then combine your point estimate (p̂) with a margin of error.

Confidence Interval for a Population Proportion Summary

A confidence interval for a population proportion estimates an unknown population proportion, p, using a sample proportion, p̂. The AP Stats procedure is a one-sample z-interval for a proportion, and the interval follows the structure point estimate ± margin of error.

Before calculating, check the conditions: random sample or randomized experiment, independence with the 10% condition when sampling without replacement, and large counts using np̂ ≥ 10 and n(1−p̂) ≥ 10. Then calculate p̂ ± z* √(p̂(1−p̂)/n) and interpret the interval in context.

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Why This Matters for the AP Statistics Exam

Confidence intervals for proportions show up on both multiple-choice and free-response questions. You need to recognize when a one-sample z-interval for a proportion is the right procedure, verify conditions, calculate the interval correctly, and write it with the proper units and context. Showing your name of procedure, condition checks, and clear setup is important for clear exam work, since graders look for evidence that you chose and applied the right method.

This topic also sets up later skills. Once you can build and read a proportion interval, you can justify claims (6.3), connect intervals to hypothesis tests, and extend the same logic to differences between two proportions (6.8).

Key Takeaways

• The correct procedure for one categorical variable from one sample is a one-sample z-interval for a proportion.
• The interval has the form point estimate ± margin of error, or p̂ ± z* √(p̂(1−p̂)/n).
• Check three things before calculating: data come from a random sample or randomized experiment, the 10% condition (n ≤ 0.10N) when sampling without replacement, and the large counts condition (np̂ ≥ 10 and n(1−p̂) ≥ 10).
• The margin of error is z* · SE, where the standard error is √(p̂(1−p̂)/n).
• Higher confidence levels make the interval wider; larger sample sizes make it narrower (width is proportional to 1/√n).
• To find a minimum sample size for a target margin of error, rearrange the formula and use a guess for p̂ or p̂ = 0.5 for the largest (safest) sample size.

What a Confidence Interval Tells You

A confidence interval is a range of values calculated from sample data that estimates a population parameter. For categorical data, that parameter is a population proportion p.

The interval is built from your sample proportion, your sample size, and the sampling distribution of p̂. The sampling distribution is the distribution of p̂ you would see if you took many samples of the same size from the population.

The confidence level (often 95%) describes how the method behaves in repeated sampling. As the confidence level increases, the interval gets wider, because you are asking for a method that captures the true proportion more often.

The Right Procedure: One-Sample z-Interval for a Proportion

When you have one categorical variable and one sample, the correct procedure is a one-sample z-interval for a proportion. This uses the normal approximation to the sampling distribution of p̂ to turn your sample proportion into a range of plausible values for the true proportion.

Checking Conditions

Always verify these before you calculate. On the exam, point to the specific words in the problem that justify each condition.

Random Sample or Randomized Experiment

Data should come from a random sample or a randomized experiment. This reduces bias from a bad sample. Quote or reference the part of the problem that states the randomness. Without it, your scope of inference breaks down, and no calculation can fix a biased sample.

Independence (10% Condition)

When sampling without replacement, check that n ≤ 0.10N, where N is the population size. This keeps each selection from meaningfully affecting the others.

For example, if you have a random sample of 85 teenagers and you are estimating the proportion of all teenagers who pass their math class, you could write: "It is reasonable to believe there are at least 850 teenagers currently enrolled in a math class." A clean way to state this is: "It is reasonable to believe our population is at least 10n."

Normal (Large Counts Condition)

Check that both the number of successes np̂ and the number of failures n(1−p̂) are at least 10. This is the large counts condition, and it confirms the sampling distribution of p̂ is approximately normal.

Using the example above, suppose 75% of the sample passes. With n = 85: 0.75(85) = 63.75, which is at least 10, and 0.25(85) = 21.25, which is also at least 10. Since both are at least 10, the sampling distribution of p̂ is approximately normal.

Calculating the Interval

A confidence interval is built from two pieces: your point estimate and your margin of error.

Point Estimate

The point estimate is the sample proportion, p̂. This is your single best estimate of the population proportion from the sample data, and it sits at the center of the interval. The bounds come from adding and subtracting the margin of error.

Margin of Error

The margin of error is the buffer you add and subtract to account for uncertainty. It is the critical value times the standard error:

Margin of error = z* · SE, where the standard error is √(p̂(1−p̂)/n).

The critical value (z*) is set by the confidence level and marks the boundaries of the middle C% of the standard normal distribution. For a 95% interval, z* is about 1.96.

Sample size matters here. As n increases, the standard error decreases, which shrinks the margin of error. So a larger sample gives a narrower interval and a more precise estimate.

Formula

The full interval is:

p̂ ± z* √(p̂(1−p̂)/n)

These interval formulas are not printed on the AP Statistics formula sheet, but you do not need to memorize them. You can build them from the general structure of point estimate ± (critical value)(standard error), using the standard error formulas that are provided.

Solving for Sample Size

The margin of error formula can be rearranged to find the minimum sample size for a target margin of error:

n = (z* / margin of error)^2 · p(1 − p)

Here p is your planning value for the proportion. If you do not have a reasonable guess, use p = 0.5, which produces the largest required sample size and gives a safe upper bound for hitting your target margin of error.

Using a Calculator

A graphing calculator makes this faster. On a TI-84, choose 1-PropZInt from the STAT TESTS menu, enter your number of successes (x), sample size (n), and confidence level, then calculate to get the interval.

Even when you use technology, write out your procedure name, condition checks, and the interval with context. That setup is important for clear exam work.

How to Use This on the AP Statistics Exam

MCQ

• Recognize when a question calls for a one-sample z-interval for a proportion (one categorical variable, one sample).
• Predict how changing the confidence level or sample size affects width and margin of error. Higher confidence means wider; larger n means narrower.
• Use the relationship that interval width is exactly twice the margin of error.

Free Response

• State the procedure: "one-sample z-interval for a proportion."
• Check all conditions and reference the problem's wording for randomness and the 10% condition, and show the large counts numbers.
• Show the formula with values plugged in, then report the interval.
• Interpret with context and units when asked, naming the sample and the population it represents.

Common Trap

• Do not skip condition checks just because you used a calculator. Graders want the named procedure and the verified conditions.

Common Misconceptions

• "95% confidence means there is a 95% chance the true proportion is in this one interval." The true proportion is fixed. The confidence level describes the method: in repeated sampling, about 95% of the intervals built this way would capture the true proportion.
• "You check large counts with the null value p₀." For a confidence interval you use the sample proportion p̂ in np̂ and n(1−p̂). The null value p₀ is used when setting up a significance test, not a confidence interval.
• "A bigger sample fixes a biased sample." Increasing n narrows the interval but does nothing to remove bias from a non-random sample. Random collection is what protects your scope of inference.
• "The margin of error and the interval width are the same thing." The width of the interval is exactly twice the margin of error.
• "You need n > 30 for proportions." The condition for a proportion interval is large counts (np̂ ≥ 10 and n(1−p̂) ≥ 10), not a fixed cutoff like 30.
• "The interval estimates the sample proportion." The interval estimates the unknown population proportion. The sample proportion is just the center of the interval.

Related AP Statistics Guides

• Unit 6 Overview: Inference for Categorical Data: Proportions
• 6.1 Introducing Statistics: Why Be Normal?
• 6.4 Setting Up a Test for a Population Proportion
• 6.3 Justifying a Claim Based on a Confidence Interval for a Population Proportion
• 6.5 Interpreting p-Values
• 6.8 Confidence Intervals for the Difference of Two Proportions