AP® Statistics review sheet from Aim for Five (aimforfive.com/stats/units/3/3-11)
Unit 3 · Topic 3.11
3.11 Justifying a Claim Based on a Confidence Interval for the Difference Between Two Population Proportions
An interval for p₁ − p₂ tells you how big the difference between two groups plausibly is, and whether "no difference" is still believable. The key question is whether 0 is in the interval.
Key terms
- interpreting the interval
- confidence level
- 0 inside the interval
- plausible values
Interpreting the interval
Template: "We are C% confident that the interval from a to b captures the true difference in [response] between [group 1] and [group 2] ([group 1] minus [group 2])."
The confidence level means: in repeated random sampling (or repeated random assignment) with the same sample sizes, about C% of intervals built this way would capture the true difference p₁ − p₂.
Any one interval either captures the true difference or it doesn't. Because it comes from samples, it could miss, which is why the conclusion is stated with confidence, not certainty.
Is 0 in the interval?
- Every value positive (like 0.013 to 0.187): convincing evidence that p₁ > p₂.
- Every value negative: convincing evidence that p₁ < p₂.
- 0 inside (like −0.04 to 0.09): a difference of 0 is plausible, so there isn't convincing evidence of a difference. Don't conclude the proportions are equal.
Size, not just direction
The interval also tells you how big the difference might be. In the flu-shot example, (0.013, 0.187) says the reminder plausibly raises the vaccination rate by anywhere from about 1 to about 19 percentage points. A 1-point increase might not be worth the cost; a 19-point increase would be. An interval near 0 can be statistically convincing and still practically small.
Testing a claimed difference
You can check any claimed difference, not just 0. If a company says its reminder raises vaccination rates by at least 5 percentage points, look at whether 0.05 is plausible. With (0.013, 0.187), it is: 0.05 is inside. But so are values below 0.05, so the interval doesn't confirm "at least 5 points" either. It only says that claim isn't contradicted.
Cause and effect, and scope
If the data come from a randomized experiment and 0 is outside the interval, you can conclude the treatment caused the difference, for subjects like those in the study. If they come from two random samples, you can generalize to the two populations, but you can't claim cause and effect.
Interpretations to avoid
- "There's a 95% chance the difference is between 0.013 and 0.187." Use "95% confident."
- "The reminder group was 1.3% to 18.7% more likely to get the shot." The interval is about the true difference, not the sample, which had a difference of exactly 0.10.
- Talking about proportions without saying which group is higher. Always connect the sign to the groups.
Worked examples
Try each one yourself first, then open the solution.
- Example 1Calculator allowed
Use the interval to make a claim
The 95% interval for the difference (reminder minus no reminder) in flu-shot rates was (0.013, 0.187), from a randomized experiment. Does it give convincing evidence that text reminders increase vaccination rates? Explain.
Show the solutionHide the solution
- Step 1: Every value in the interval is positive, so 0 is not a plausible value for p₁ − p₂.
- Step 2: That gives convincing evidence that the true proportion vaccinated is higher with a reminder.
- Step 3: Treatments were randomly assigned, so the reminder caused the increase, for patients similar to those in the study.
Answer: Yes. All plausible differences are positive (0.013 to 0.187), so there's convincing evidence that text reminders increase the proportion of patients like these who get a flu shot.
- Example 2Calculator allowed
Trap: 0 in the interval means "no difference"?
A 95% interval for the difference in the proportion of left-handed people (men minus women) is (−0.02, 0.05). A student concludes, "Men and women are equally likely to be left-handed." Correct the conclusion.
Show the solutionHide the solution
- Step 1: 0 is inside the interval, so "no difference" is plausible.
- Step 2: But so are differences as large as 0.05 or as small as −0.02.
- Step 3: You can only say there isn't convincing evidence of a difference.
Answer: The data don't provide convincing evidence of a difference, but they don't show the proportions are equal; differences from −0.02 to 0.05 are all plausible.
Common mistakes
- Concluding the proportions are equal when 0 is in the interval.
- Forgetting the order of subtraction, so the direction of the conclusion is backward.
- Claiming cause and effect from two random samples without random assignment.
On the exam
- "Does the interval support the claim?" Answer by saying where 0 (or the claimed difference) falls relative to the interval, then connect that to the claim in context.
- Mention scope: random assignment allows causation; random sampling allows generalizing.
Connected topics
- 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 33.13 Carrying Out a Test for the Difference Between Two Population Proportions
- Unit 44.8 Justifying a Claim Based on a Confidence Interval for the Difference Between Two Population Means
Videos
Check yourself
2 questions on 3.11 Justifying a Claim Based on a Confidence Interval for the Difference Between Two Population Proportions. Pick an answer to see if you got it, and why.
A 95% confidence interval for p₁ − p₂, the difference between the proportions of adults in two states who exercise daily, is (−0.032, 0.058). Which conclusion is appropriate?
A clinic randomly assigned 490 patients with upcoming appointments to two groups. The 250 patients in one group got a text reminder the day before, and 185 of them showed up. The 240 patients in the other group got no reminder, and 156 of them showed up.
Described experiment with invented results
Based on the interval (0.009, 0.171), what can the clinic conclude?
0 of 2 answered