AP Statistics · Unit 3 · Topic 3.3

Confidence Interval for a Proportion Study Lab

A confidence interval gives a range of plausible values for a population proportion: point estimate ± margin of error. Build one from sample data, check the conditions, and then watch hundreds of simulated intervals to see what the confidence level really promises.

1. Build a one-proportion z-interval

Enter the sample results and pick a confidence level. Each step of the interval is worked out below.

p̂ ± z* · √( p̂(1 − p̂) ÷ n ) ME = z* · SE

Preset studies

Successes x

Sample size n

Population size NFor the 10% condition

How was the sample chosen?

Confidence level

The interval

Middle C% of the modelp̂ (point estimate)

p̂

Critical value z*

Margin of error

Interval

Conditions

Show your work

What does "95% confident" mean? Capture the true p

Pretend you know the true p. Take many random samples of size n (from above) and build an interval from each one. About C% of them should capture p.

True p

Intervals made

Captured p

Last 40 intervals red ones missed

Planning: how big a sample?

Solve z* · √(p*(1 − p*)/n) ≤ ME for n. With no prior guess, use p* = 0.5, which gives the largest (safest) n.

Desired margin of error

Prior guess p*

Uses the confidence level chosen above.

Sample size work

2. Key ideas

Structure

Estimate ± margin of error

Every confidence interval in AP Statistics is statistic ± (critical value) × (standard error of the statistic).

Standard error

SE uses p̂

We don't know p, so we estimate σ_p̂ with SE = √(p̂(1 − p̂)/n).

Critical value

z* from the confidence level

z* cuts off the middle C% of the standard normal: 1.645 (90%), 1.96 (95%), 2.576 (99%).

Conditions

Random, 10%, Large counts

Random sample; n ≤ 10% of the population when sampling without replacement; at least 10 successes and 10 failures in the sample.

Width

What changes the margin?

Higher confidence → wider. Larger n → narrower (4× the n gives half the width). p̂ near 0.5 → widest.

Confidence level

It describes the method

About C% of intervals from repeated random samples capture p. A single interval either does or doesn't; we never know which.

3. Quick reference

Common confidence levels and how the margin of error responds.

Confidence level	z*	Tail area each side	ME when p̂ = 0.5, n = 400
80%	1.282	0.10	0.032
90%	1.645	0.05	0.041
95%	1.960	0.025	0.049
98%	2.326	0.01	0.058
99%	2.576	0.005	0.064

4. On the FRQ

How to earn the points

Use the four-step process: State (parameter and confidence level), Plan (name the procedure, check conditions), Do (show the formula with numbers), Conclude (interpret in context).
Define the parameter in words: "p = the true proportion of all students at the school who feel well rested."
Check conditions with numbers: "np̂ = 58 ≥ 10 and n(1 − p̂) = 142 ≥ 10; 200 < 10% of 2,400 students."
Name the procedure: "one-sample z-interval for p."
Interpret: "We are 95% confident that the interval from 0.227 to 0.353 captures the true proportion of all students at the school who feel well rested."

Common mistakes

Saying "there is a 95% probability that p is in this interval."
Interpreting the interval as describing sample proportions instead of the population proportion.
Using p instead of p̂ in the standard error, or checking Large Counts with a p you don't know.
Rounding a required sample size down. Always round up.
Thinking a bigger sample fixes a biased (non-random) sampling method.

5. Practice questions

AP-style multiple choice. Pick an answer to see the explanation.

0 / 0 correct

6. Key terms

The vocabulary for topic 3.3. Read the list, flip through flashcards, or quiz yourself. For every term in the unit, see the Unit 3 terminology guide.