Short on time? This page is the recap. For explanations and worked examples, use the full study guide.
Recap
- μ₁ − μ₂ is fixed; the interval changes from sample to sample. One interval may or may not contain μ₁ − μ₂.
- Interval template: “We are C% confident that the interval from a to b [units] captures the difference (group 1 minus group 2) in the mean [response] of all [population 1] and all [population 2].”
- Confidence level template: “In repeated random sampling with the same sample sizes from these populations, about C% of intervals built this way would capture μ₁ − μ₂.”
- For an experiment, refer to the true difference in mean response between the treatments for subjects like these.
Key relationships
| Situation | Conclusion |
|---|---|
| Both limits positive | Convincing evidence μ₁ > μ₂ |
| Both limits negative | Convincing evidence μ₁ < μ₂ |
| Interval contains 0 | No convincing evidence of a difference (not proof of equal means) |
| Claimed value (or boundary) outside the interval | Convincing evidence against “μ₁ − μ₂ equals it” |
| Every value in the interval satisfies a directional claim | Convincing evidence for the claim |
| Interval straddles the claim’s boundary | Not convincing either way |
| Reverse order of subtraction | (a, b) becomes (−b, −a); same conclusion |
| Higher confidence level | Larger t*, wider interval |
Assumptions
- The conditions for the two-sample t-interval were met (Topic 4.7). Otherwise, no claim can be justified.
- Random samples allow generalising to the populations; random assignment allows cause-and-effect conclusions.
Mistakes to avoid
- “Contains 0, so the means are equal.”
- “Mostly positive, so group 1 is higher.” Look at whether 0 is inside, not how much of the interval is positive.
- “95% probability” for one computed interval.
- Interpreting sample means or individual values instead of population means.
- No order of subtraction, so the sign cannot be read.
- Cause and effect from an observational study.
- A conclusion with no reference to the interval.
Quick self-check
- A 95% interval for μ₁ − μ₂ is (−4.1, 2.7). Is there convincing evidence of a difference? (No: 0 is inside the interval, so 0 is a plausible difference.)
- An interval for μ_A − μ_B is (1.2, 6.8). Give the interval for μ_B − μ_A. ((−6.8, −1.2))
- An interval for μ₁ − μ₂ is (3.5, 9.5). Judge the claims (i) “group 1’s mean is at least 2 higher” and (ii) “group 1’s mean is more than 4 higher”. ((i) Convincing evidence for: every value exceeds 2. (ii) Not convincing: values from 3.5 to 4 are plausible.)
Next: practice questions.