AP Statistics · Unit 3 · Topic 3.4

Justifying a Claim with a Proportion Interval Study Lab

A confidence interval gives the plausible values for a population proportion. If a claimed value is outside the interval, the data give convincing evidence against it. If it is inside, the claim is plausible, but not proven. Move the interval and the claim to see the logic.

1. Does the interval support the claim?

A confidence interval is the set of plausible values for p. To judge a claim, look at where the claimed value sits relative to the interval.

Preset claims

Sample proportion p̂

Sample size n

Claimed value

Claim

Confidence

Interval vs. claim

Confidence interval (plausible values)Claimed valueValues the claim allows

Interval

Margin of error

Claimed value inside?

Verdict

Show your work

Judge the statement

A random sample of 1,000 adults gives a 95% confidence interval of (0.43, 0.49) for the proportion of all adults in the state who support a new park tax. Is each statement correct?

2. Key ideas

Plausible values

Inside means plausible

Every value inside the interval is consistent with the data. Values outside it are not plausible at that confidence level.

Equals claims

p = value

If the value is outside the interval, there is convincing evidence p differs from it. If inside, we can't rule it out.

Direction claims

p > value or p < value

For convincing evidence that p > 0.5, the whole interval must be above 0.5. If the interval contains 0.5, the evidence is not convincing.

Not proof

Plausible ≠ true

A plausible claim isn't proven. Other values in the interval are equally plausible, and the interval itself might miss p.

Width matters

Confidence and n

Wider intervals (higher confidence, smaller n) contain more plausible values, so they rule out fewer claims.

Link to tests

Same decision, two tools

A C% two-sided interval and a two-sided test at α = 1 − C usually agree on whether a claimed value is plausible.

3. Quick reference

How to word your conclusion from a confidence interval.

Claim	What the interval shows	Conclusion
p = 0.50	0.50 inside the interval	Plausible; no convincing evidence against the claim
p = 0.50	0.50 outside the interval	Convincing evidence that p ≠ 0.50
p > 0.50	Entire interval above 0.50	Convincing evidence that p > 0.50
p > 0.50	Interval includes 0.50 or values below	Not convincing; values ≤ 0.50 are plausible
p < 0.30	Entire interval below 0.30	Convincing evidence that p < 0.30

4. On the FRQ

How to earn the points

State the interval and the claimed value, then say where the value falls: "Because 0.50 is not in the interval (0.38, 0.46)…"
Use the word "plausible" and connect to the claim: "…0.50 is not a plausible value for the true proportion of residents who support the stadium."
For a directional claim, say "all values in the interval are greater than 0.50."
Avoid "proves" or "accept." Say "does not provide convincing evidence against the claim."
Keep the conclusion about the population parameter, in context.

Common mistakes

Concluding the claim is true because the value is in the interval. It is only plausible.
Using p̂ alone to judge a claim ("p̂ = 0.53 > 0.50, so a majority") without accounting for the margin of error.
Saying a value just outside the interval is plausible because it is "close."
Forgetting that a directional claim requires the entire interval on one side.
Talking about the sample ("53% of the sample") instead of the population in the conclusion.

5. Practice questions

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

0 / 0 correct

6. Key terms

The vocabulary for topic 3.4. Read the list, flip through flashcards, or quiz yourself. For every term in the unit, see the Unit 3 terminology guide.