Once you have a confidence interval for μ₁ − μ₂, you can use it to judge a claim. Is a difference of 0 plausible? Is the claimed advantage convincingly supported? Set a claim, build the interval, and read the verdict. Then explore how confidence level, sample size, and variability change the interval's width.
A confidence interval lists the plausible values of μ₁ − μ₂. To judge a claim, check where the claimed value sits: inside the interval means it is plausible; outside means the data give convincing evidence against it.
A narrow interval pins down μ₁ − μ₂ more precisely. Here both groups share one sample size n and one standard deviation s, so you can see each effect on the margin of error.
Every value in the interval is a plausible value of μ₁ − μ₂. Values outside the interval are not plausible at that confidence level.
If 0 is outside the interval, the data give convincing evidence that the population means differ. If 0 is inside, "no difference" is plausible, so we do not have convincing evidence of a difference.
When 0 is in the interval, never say the means are equal. Many other values, some positive and some negative, are plausible too.
For "group 1's mean is more than d higher," you need every plausible value above d, so the lower endpoint must exceed d. If d is inside the interval, the claim is plausible but not convincingly supported.
The interval gets narrower with a lower confidence level, larger samples, or less variability in the data. Quadrupling both sample sizes roughly halves the margin of error.
A C% interval and a two-sided test at α = 1 − C give the same answer about d: d is rejected by the test exactly when it is outside the interval.
How to word the conclusion for each situation. Here the claimed value is d.
| Claim | Interval | Conclusion |
|---|---|---|
| The means are different (d = 0) | (2.1, 7.4), all above 0 | Convincing evidence of a difference; μ₁ is likely larger |
| The means are different (d = 0) | (−3.2, 1.5), contains 0 | No convincing evidence of a difference; 0 is plausible |
| Group 1's mean is greater (d = 0) | (−8.0, −1.1), all below 0 | Convincing evidence against the claim; μ₁ is likely smaller |
| Group 1's mean is more than 3 higher (d = 3) | (1.8, 6.2), contains 3 | Plausible, but not convincing; values below 3 are also plausible |
| Group 1's mean is more than 3 higher (d = 3) | (3.4, 6.2), all above 3 | Convincing evidence for the claim |
AP-style multiple choice. Pick an answer to see the explanation.
The vocabulary for topic 4.8. Read the list, flip through flashcards, or quiz yourself. For every term in the unit, see the Unit 4 terminology guide.