AP® Statistics review sheet from Aim for Five (aimforfive.com/stats/units/3/3-4)
Unit 3 · Topic 3.4
3.4 Justifying a Claim Based on a Confidence Interval for a Population Proportion
Building an interval is half the job; you also have to interpret it and use it. This topic covers what "95% confident" means (and doesn't mean), how to use an interval to judge a claim and how confidence level and sample size change the width.
Key terms
- interpreting a confidence interval
- confidence level
- plausible values
- interval width
Interpreting the interval
Template: "We are C% confident that the interval from a to b captures the true proportion of [population] who [response]." For example: "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."
Any one interval either contains p or it doesn't. You can't know which. That's why you say "confident," not "there's a 95% probability that p is in this interval."
Interpreting the confidence level
The confidence level describes the method, not one interval. "If we took many random samples of 400 adults from this city and built a 95% interval from each, about 95% of those intervals would capture the true proportion who read news daily."
Think of it as a success rate for the process. Each new sample gives a different p̂ and a different interval; about 95% of them catch p and about 5% miss.
Using an interval to judge a claim
Values inside the interval are plausible values of p. Values outside are not plausible at that confidence level.
If a news site claims 40% of the city's adults read news daily, and your interval is (0.284, 0.376), the claim isn't plausible: 0.40 is above the whole interval. If someone claims 30%, that's plausible, because 0.30 is inside. Plausible doesn't mean proven; other values in the interval are equally plausible.
What changes the width
The trade-off: for a fixed sample, you can only gain confidence by giving up precision. To get both, take a bigger sample.
- Higher confidence level: bigger z*, bigger margin of error, wider interval. Going from 95% to 99% confidence widens the interval.
- Larger sample size: smaller SE, narrower interval. The width is roughly proportional to 1/√n, so quadrupling n roughly halves the width.
- The value of p̂: intervals are widest when p̂ is near 0.5.
Interpretations to avoid
- "There's a 95% probability that p is between 0.284 and 0.376." The probability language belongs to the method, not one interval.
- "95% of adults read news between 28.4% and 37.6% of the time." The interval is about one proportion, not about individuals.
- "95% of samples will have p̂ between 0.284 and 0.376." Different samples give different intervals centered at their own p̂.
- "We are 95% confident the sample proportion is in the interval." p̂ is the center of the interval; it's always there.
Worked examples
Try each one yourself first, then open the solution.
- Example 1Calculator allowed
Use an interval to evaluate claims
A 95% confidence interval for the proportion of all students at a large high school who have a part-time job is (0.21, 0.33). The principal claims that one-quarter of students work. A parent claims more than a third do. Evaluate both claims.
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- Step 1: One-quarter is 0.25. It's inside (0.21, 0.33), so it's a plausible value. The interval doesn't give evidence against the principal.
- Step 2: More than a third means p > 0.333. Every value in the interval is below 0.333, so the parent's claim isn't plausible at the 95% level.
- Step 3: Context: these are statements about all students at the school, not just the sample.
Answer: The principal's claim (0.25) is plausible because it's in the interval. The parent's claim is not supported: the entire interval is below 1/3.
- Example 2Calculator allowed
Effects of confidence level and sample size
The 95% interval from 3.3 was (0.284, 0.376) with n = 400 and margin of error 0.046. (a) What is the 90% interval? (b) About what margin of error would a 95% interval have with n = 1,600 and the same p̂?
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- Step 1: (a) z* = 1.645. Margin of error = 1.645 × 0.0235 ≈ 0.039. Interval: 0.33 ± 0.039 = (0.291, 0.369). It's narrower, since lower confidence uses a smaller z*.
- Step 2: (b) 1,600 is 4 times 400, so SE is divided by √4 = 2. The margin of error is about 0.046/2 = 0.023.
Answer: (a) About (0.291, 0.369). (b) About 0.023, half the original.
- Example 3Calculator allowed
Trap: a probability statement about one interval
A student writes: "There is a 95% chance that the true proportion is between 0.284 and 0.376." What's wrong, and how should it be fixed?
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- Step 1: Once the interval is computed, p is either in it or not; nothing is random any more.
- Step 2: The 95% describes how often the method works over many samples.
- Step 3: Fix: "We are 95% confident that the interval from 0.284 to 0.376 captures the true proportion…"
Answer: The 95% is about the long-run success rate of the method, not a probability for this one interval. Use "95% confident."
Common mistakes
- Saying "95% of the population" or "95% of the sample" falls in the interval. The interval estimates a single parameter.
- Interpreting the interval about p̂ ("captures the sample proportion"). p̂ is always at the center by construction.
- Thinking a higher confidence level gives a narrower interval.
- Treating a value inside the interval as proven true.
On the exam
- Free-response questions often ask for both the interval interpretation and the confidence level interpretation. They're different; practice both templates.
- To use an interval to support a claim, say whether the claimed value is inside or outside the interval and connect that to plausibility, in context.
Connected topics
- Unit 33.3 Constructing a Confidence Interval for a Population Proportion
- Unit 33.7 Carrying Out a Test for a Population Proportion
- Unit 33.11 Justifying a Claim Based on a Confidence Interval for the Difference Between Two Population Proportions
- Unit 44.3 Justifying a Claim Based on a Confidence Interval for a Population Mean or Population Mean Difference
Videos
Check yourself
4 questions on 3.4 Justifying a Claim Based on a Confidence Interval for a Population Proportion. Pick an answer to see if you got it, and why.
A researcher builds a 90% confidence interval for a population proportion. What does "90% confidence" mean?
A 95% confidence interval for a population proportion is (0.31, 0.39). Using the same data, which could be a 99% confidence interval?
A student plans to build a 95% confidence interval for the proportion of students at her school who have a part-time job. She switches from a random sample of 100 students to a random sample of 400. If p̂ stays about the same, how will the margin of error change?
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?
0 of 4 answered