AP Statistics Confidence Intervals for a Proportion Lesson (Free PDF)
This confidence interval proportion lesson for AP Statistics covers confidence intervals for a proportion (Common Core AP STAT 6.1, AP STAT 6.2, AP STAT 6.3, HSS-IC.B.4), with a free PDF to print.
A confidence interval for a proportion has the form p-hat plus or minus z* times the square root of p-hat(1-p-hat)/n. Check the random, 10 percent, and large counts conditions, then interpret: we are C percent confident the interval captures the true proportion. The confidence level is the long-run percent of such intervals that capture the parameter.
AP Statistics · Common Core State Standards: AP STAT 6.1, AP STAT 6.2, AP STAT 6.3, HSS-IC.B.4
Key facts in this confidence interval proportion lesson
- Interval: p-hat +/- z* sqrt(p-hat(1 – p-hat)/n); z* = 1.645, 1.960, 2.576 for 90%, 95%, 99%.
- Conditions: Random, 10% (n < 10% of population), Large counts (n p-hat and n(1 - p-hat) at least 10).
- Interpret the interval in context; interpret the level as the long-run capture rate.
- Higher confidence or smaller n gives a wider interval; quadrupling n halves the margin of error.
- A claimed value outside the interval is not plausible at that confidence level.
What is a confidence interval?
A confidence interval is a range of plausible values for a population parameter, built from a sample statistic. For a population proportion \(p\), the one-sample \(z\) interval is:
\[\hat{p}\pm z^{*}\sqrt{\frac{\hat{p}(1-\hat{p})}{n}}\]
The quantity \(z^{*}\sqrt{\hat{p}(1-\hat{p})/n}\) is the margin of error. The value \(z^{*}\) is the critical value for the chosen confidence level.
How do you build and check the interval?
Example: in a random sample of \(n=600\) students, \(x=312\) ride the bus, so \(\hat{p}=\frac{312}{600}=0.52\).
- Random: the students were selected at random.
- 10%: 600 is less than 10% of the school.
- Large counts: \(n\hat{p}=312\) and \(n(1-\hat{p})=288\) are both at least 10.
Standard error: \(\sqrt{\frac{0.52(0.48)}{600}}=0.0204\). Margin of error: \(1.96(0.0204)=0.0400\). Interval: \((0.480,\ 0.560)\).
The conditions matter because the formula assumes that p-hat varies approximately normally from sample to sample. A random sample guards against bias, the 10% condition makes the sample members close to independent, and the large counts condition makes the sampling distribution of p-hat bell-shaped. If any condition fails, the stated confidence level may not be accurate, and you should say so in your answer. Always show each condition with the numbers from the problem rather than writing only that it is satisfied.
How do you interpret the interval and the confidence level?
Interval: We are 95% confident that the interval from 0.480 to 0.560 captures the true proportion of all students who ride the bus.
Level: If we took many random samples and made a 95% interval from each, about 95% of those intervals would capture the true proportion. The confidence level describes the method, not any single interval.
Wider intervals come from higher confidence or smaller samples. To cut the margin of error in half, multiply the sample size by 4. If a claimed value (such as 0.5) lies inside the interval, it is plausible; if it lies outside, the data provide evidence against it.
Confidence interval proportion lesson: questions and answers
What does 95% confidence mean?
It means the method captures the true parameter in about 95% of all random samples. It is not the probability that this one fixed interval contains the parameter; the parameter is fixed and the interval is random.
What are the conditions for a z interval for a proportion?
The data must come from a random sample or random assignment, the sample must be less than 10% of the population, and the number of successes and failures in the sample must each be at least 10.
How does sample size affect the margin of error?
The margin of error is proportional to 1 over the square root of n. To halve it, you need four times as many observations.
How do I find the sample size for a desired margin of error?
Solve n >= (z*/ME)^2 p*(1 – p*). If no estimate of p is available, use p* = 0.5, which gives the largest, safest sample size.
Can a confidence interval be used to test a claim?
Yes. If the claimed proportion lies inside the interval, the claim is plausible; if it lies outside, the interval gives convincing evidence against the claim at that confidence level.
How to use this confidence interval proportion lesson
Read the definitions first, then follow the worked method line by line and redo the example on paper without looking. When every step makes sense, move on to the worksheet, then check yourself with the quiz.
Download the confidence interval proportion lesson as a PDF with the button below and keep it with your notes.
Download the confidence interval proportion lesson as a PDF with the button below and keep it with your notes.
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