AP® Statistics review sheet from Aim for Five (aimforfive.com/stats/units/3/3-3)
Unit 3 · Topic 3.3
3.3 Constructing a Confidence Interval for a Population Proportion
A confidence interval gives a range of plausible values for a population proportion instead of a single guess. This topic covers the one-sample z-interval: naming the parameter, checking conditions, computing p̂ ± z*·SE and finding the sample size needed for a target margin of error.
Key terms
- confidence interval
- one-sample z-interval
- critical value z*
- standard error (SE)
- margin of error
- sample size for a margin of error
The structure of an interval
Every confidence interval in this course has the form statistic ± (critical value)(standard error). For one proportion: p̂ ± z* √(p̂(1 − p̂)/n).
The standard error SE = √(p̂(1 − p̂)/n) estimates the standard deviation of the sampling distribution. It uses p̂ because the true p is unknown. It tells you how far p̂ typically is from p.
The margin of error is z* × SE. It's half the width of the interval.
Critical values
z* is chosen so that C% of the area under the standard normal curve sits between −z* and z*. Find it with invNorm or from Table A: for 95%, the area to the left of z* is 0.975.
| Confidence level | z* |
|---|---|
| 90% | 1.645 |
| 95% | 1.960 |
| 99% | 2.576 |
Conditions
Check each condition with numbers and context. "Random: the problem says a random sample of 400 adults was selected" is better than writing "SRS" alone.
- Random: the data come from a random sample from the population (name it in context).
- 10%: when sampling without replacement, the sample is no more than 10% of the population (n ≤ 0.10N).
- Normal: the observed number of successes np̂ and failures n(1 − p̂) are both at least 10.
A full interval, step by step
A complete answer has four parts. Name the procedure and parameter: "one-sample z-interval for p, the true proportion of all adults in the city who read news daily." Check the conditions. Do the math with the formula shown. Interpret in context (topic 3.4).
Choosing a sample size
To get a margin of error no bigger than M, solve z*√(p̂(1 − p̂)/n) ≤ M for n: n ≥ (z*/M)² · p̂(1 − p̂). If you have no estimate of p, use p̂ = 0.5. That makes p̂(1 − p̂) as large as possible, so the sample size is big enough no matter what p is.
Always round the answer up to the next whole number. Rounding down would give a margin of error slightly too big.
What the margin of error covers
The margin of error only accounts for random sampling variability, the chance differences between one random sample and another. It does not cover bias from undercoverage, nonresponse or bad question wording. A poll with a small margin of error can still be far off if the sample was badly collected.
On a calculator, 1-PropZInt does the arithmetic. On free response, still show the formula with values plugged in, then the interval.
Worked examples
Try each one yourself first, then open the solution.
- Example 1Calculator allowed
Construct a 95% interval
In a random sample of 400 adults in a large city, 132 say they read news every day. Construct a 95% confidence interval for the proportion of all adults in the city who read news daily.
Show the solutionHide the solution
- Step 1: Procedure: one-sample z-interval for p = the true proportion of all adults in the city who read news daily.
- Step 2: Random: random sample of 400 adults. 10%: 400 is less than 10% of all adults in a large city. Normal: np̂ = 132 and n(1 − p̂) = 268, both at least 10.
- Step 3: p̂ = 132/400 = 0.33. SE = √(0.33 × 0.67/400) ≈ 0.0235.
- Step 4: Margin of error: 1.96 × 0.0235 ≈ 0.046.
- Step 5: Interval: 0.33 ± 0.046, or (0.284, 0.376).
Answer: (0.284, 0.376). We are 95% confident that the interval from 0.284 to 0.376 captures the true proportion of all adults in the city who read news daily.
- Example 2Calculator allowed
Sample size for a margin of error
A researcher wants a 95% confidence interval for a proportion with a margin of error of at most 0.03. (a) With no prior estimate, how many people should she sample? (b) If an earlier study found p̂ = 0.33?
Show the solutionHide the solution
- Step 1: (a) Use p̂ = 0.5: n ≥ (1.96/0.03)² × 0.5 × 0.5 ≈ 1,067.1. Round up: 1,068.
- Step 2: (b) Use p̂ = 0.33: n ≥ (1.96/0.03)² × 0.33 × 0.67 ≈ 943.8. Round up: 944.
- Step 3: The prior estimate lowers the required sample, because 0.33 × 0.67 is less than 0.25.
Answer: (a) 1,068 people. (b) 944 people.
- Example 3Calculator allowed
Trap: rounding the sample size down
A student computes n ≥ 1,067.1 and reports 1,067. What's wrong?
Show the solutionHide the solution
- Step 1: With n = 1,067, the margin of error would be slightly larger than 0.03.
- Step 2: The requirement is "at most 0.03," so you need at least 1,067.1 people, which means 1,068.
Answer: Always round sample sizes up: 1,068.
Common mistakes
- Checking the normal condition with n ≥ 30 instead of np̂ ≥ 10 and n(1 − p̂) ≥ 10.
- Using z* = 1.96 for every confidence level.
- Rounding a required sample size down.
- Defining the parameter as the sample proportion ("the proportion of the 400 adults"). The parameter is about the whole population.
On the exam
- The inference free-response question rewards four things: the named procedure and parameter, conditions checked in context, correct mechanics and an interpretation. Practice writing all four every time.
- When technology gives you the interval, still write the formula with numbers substituted, so the reader sees your process.
Connected topics
- Unit 33.2 Sampling Distributions for Sample Proportions
- Unit 33.4 Justifying a Claim Based on a Confidence Interval for a Population Proportion
- Unit 33.10 Constructing a Confidence Interval for the Difference Between Two Population Proportions
- Unit 44.2 Constructing a Confidence Interval for a Population Mean or Population Mean Difference
Videos
Check yourself
5 questions on 3.3 Constructing a Confidence Interval for a Population Proportion. Pick an answer to see if you got it, and why.
What is the critical value z* for a 99% confidence interval for a population proportion?
A polling organization wants to estimate the proportion of voters who support a ballot measure to within a margin of error of 0.03 with 95% confidence. It has no prior estimate of the proportion. What is the smallest sample size that will work?
A news report says that 54% of a state's adults favor a new law, with a margin of error of 3 percentage points at 95% confidence. Which source of error does this margin of error account for?
In a random sample of 500 adults from a large city, 210 said they pay for more than one video streaming service.
Described survey with invented results
Which of the following is a 95% confidence interval for the proportion of all adults in the city who pay for more than one streaming service?
Which condition check is correct for this interval?
0 of 5 answered