AP® Statistics review sheet from Aim for Five (aimforfive.com/stats/units/4/4-8)
Unit 4 · Topic 4.8
4.8 Justifying a Claim Based on a Confidence Interval for the Difference Between Two Population Means
A two-sample t-interval gives plausible values for the difference between two population means. Interpret it in context, check whether 0 is inside it to judge a claimed difference, and keep scope in mind: random assignment allows cause and effect, random sampling allows generalizing.
Key terms
- interpreting the interval
- confidence level
- 0 inside the interval
- claim about a difference
Interpreting the interval
Template: "We are C% confident that the interval from a to b captures the true difference in mean [variable] between [group 1] and [group 2] ([group 1] minus [group 2])."
Interpreting the confidence level: "If this experiment were repeated many times with new random assignments, about 95% of the intervals built this way would capture the true difference in mean growth." For two random samples, refer to repeated random sampling instead.
Is 0 in the interval?
- All values positive, like (0.51, 6.29): convincing evidence that μ₁ > μ₂.
- All values negative: convincing evidence that μ₁ < μ₂.
- 0 inside, like (−1.8, 3.5): no convincing evidence of a difference. A difference of 0 is plausible, but so are the other values in the interval. Don't conclude the means are equal.
How big is the difference?
The interval also tells you the size of the effect. For the fertilizer study, Fertilizer A plausibly adds anywhere from about 0.5 cm to about 6.3 cm of growth over 6 weeks. If 0.5 cm wouldn't be worth the cost of switching, the result is statistically convincing but might not be practically important.
A wide interval like this one comes from small samples. Larger groups would narrow it.
A 99% interval from the same seedling data would be wider than (0.51, 6.29) and could include 0, which would change the conclusion. That's why the confidence level is chosen before the analysis, not adjusted afterward to get the answer you want.
Scope of conclusions
Randomized experiment and 0 outside the interval: the treatment caused the difference, for units like those in the study.
Two random samples and 0 outside the interval: the population means differ, but you can't say why. Observational data can't rule out confounding.
Interpretations to avoid
- "There's a 95% chance the true difference is between 0.51 and 6.29." Use "95% confident."
- "95% of A seedlings grew 0.51 to 6.29 cm more than B seedlings." The interval is about the difference in means, not individual plants.
- "The sample difference is between 0.51 and 6.29." The sample difference is exactly 3.4; the interval estimates the true difference.
Checking other claimed differences
You can test any claimed difference against the interval, not just 0. If the company claimed A adds at least 8 cm of growth on average, the interval says no: 8 is above every plausible value. If it claimed A adds about 3 cm, that's plausible.
Worked examples
Try each one yourself first, then open the solution.
- Example 1Calculator allowed
Make a claim from the interval
The 95% interval for μA − μB (mean 6-week seedling growth, Fertilizer A minus B) from a randomized experiment is (0.51, 6.29) cm. A gardening company claims Fertilizer A causes more growth. Does the interval support the claim?
Show the solutionHide the solution
- Step 1: Every value in the interval is positive, so 0 is not plausible.
- Step 2: That's convincing evidence that the true mean growth is greater with Fertilizer A.
- Step 3: Treatments were randomly assigned, so a cause-and-effect conclusion is justified for seedlings like these.
Answer: Yes. All plausible differences are positive, and because of random assignment, there's convincing evidence that Fertilizer A causes greater mean growth than B for seedlings like these.
- Example 2Calculator allowed
Trap: 0 inside means equal
Independent random samples of first-year and second-year students give a 95% interval for the difference in mean weekly study hours (first-year minus second-year) of (−1.8, 3.5). A student concludes, "First-year and second-year students study the same amount on average." Correct this.
Show the solutionHide the solution
- Step 1: 0 is in the interval, so "no difference" is plausible.
- Step 2: But differences from −1.8 to 3.5 hours are all plausible, including real differences in either direction.
- Step 3: So there's no convincing evidence of a difference, which is not evidence of no difference.
Answer: The data don't give convincing evidence that the mean study hours differ, but they don't show the means are equal either.
Common mistakes
- Concluding the means are equal when 0 is in the interval.
- Reversing the direction of the conclusion by forgetting the order of subtraction.
- Claiming cause and effect from two random samples.
- Interpreting the interval as a range for individual differences.
On the exam
- "Does the interval provide convincing evidence…?" Answer with where 0 (or the claimed value) falls, then connect to the claim in context.
- Add a scope statement when the question asks what can be concluded: cause and effect needs random assignment.
Connected topics
- 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
- Unit 44.7 Constructing a Confidence Interval for the Difference Between Two Population Means
- Unit 44.10 Carrying Out a Test for the Difference Between Two Population Means
Videos
Check yourself
4 questions on 4.8 Justifying a Claim Based on a Confidence Interval for the Difference Between Two Population Means. Pick an answer to see if you got it, and why.
A 95% confidence interval for μ₁ − μ₂, the difference in mean daily screen time (hours) between teens in two countries, is (−0.4, 1.1). Which conclusion is appropriate?
Two random samples of adults, one from City A and one from City B, give a 90% confidence interval for μ_A − μ_B of (3.2, 7.6) minutes, where μ is the mean daily commute time. Which is the best interpretation?
A 95% confidence interval for μ₁ − μ₂ is (2.3, 9.8). Based on this interval, what would happen in a two-sided test of H₀: μ₁ = μ₂ at α = 0.05 using the same data?
| Method | Number of students | Mean score | Standard deviation |
|---|---|---|---|
| A | 35 | 72.4 | 8.1 |
| B | 32 | 67.9 | 9.6 |
Invented data: final exam scores for 67 students randomly assigned to one of two teaching methods
Which conclusion about the two methods does the 95% interval support?
0 of 4 answered