Justifying a Claim about Two Means
IntroducedReverse the subtraction before changing the claim
- A valid interval for mean A minus mean B is (−4.8, −1.2) minutes. It favours a higher mean for B.
- For B minus A, negate both endpoints and reverse their order to get (1.2, 4.8). B exceeds A by a plausible 1.2 to 4.8 minutes.
An interval for A - B is (-4.8, -1.2) minutes. What is the upper endpoint for B - A?
Negate and reorder: (1.2, 4.8) minutes.
Interpreting the difference interval
- Read it as: "We're $95\%$ confident the true difference $\mu_1 - \mu_2$ is between the endpoints." Name both groups, what the mean measures, and the units, in context.
- A positive interval suggests group $1$'s mean is higher; negative, group $2$'s. It estimates how much the two means differ, with uncertainty.
Is a real difference plausible?
- The question: do the two group means genuinely differ? Look at where the interval sits relative to zero.
- The interval is a set of plausible values for $\mu_1 - \mu_2$ — possibly including $0$. "No difference" is exactly $\mu_1 - \mu_2 = 0$.
The role of zero
- Interval contains $0$ → "no difference" is plausible → not convincing evidence of a difference. Interval entirely above or below $0$ → convincing evidence the means differ.
- The position of zero is the whole decision. An all-positive or all-negative interval also gives the direction.
A 95% interval for μ1 − μ2 is (1.2, 4.8). You can conclude...
The interval is entirely positive (above 0).
A 95% interval for μ1 − μ2 is (−0.5, 3.1). You can conclude...
The interval contains 0, so 'no difference' is plausible.
For a difference-of-means interval, the value representing 'no difference' is 0.
μ1 − μ2 = 0 means equal means.
An interval that contains 0 proves the two means are exactly equal.
It only means equality can't be ruled out.
An entirely negative interval for μ1 − μ2 suggests...
μ1 − μ2 < 0 means μ2 > μ1.
Match each interval for μ1 - μ2 to the interval-based conclusion.
Read the sign of the entire interval in its defined subtraction order.
Justify the conclusion
- Tie the conclusion to the interval and to zero, in context. "Since $(1.2, 4.8)$ is entirely positive, there's convincing evidence group $1$'s mean is higher."
- Or "Since $(-0.5, 3.1)$ contains $0$, there's no convincing evidence of a difference." Always phrase it about the two population means.
Zero is the pivot for a mean-difference interval too. A captured $0$ means "no difference is plausible" — you can not conclude the means differ (and it doesn't prove them equal). This mirrors a two-sided $t$-test at the matching level. Same logic as the difference-of-proportions interval, now for means.
Interval for $\mu_1 - \mu_2$ (minutes of improvement):
- $(1.2,\ 4.8)$: entirely above $0$ → convincing evidence group $1$ improved more.
- $(-0.5,\ 3.1)$: contains $0$ → no convincing evidence of a difference.
- The captured $0$ flips the conclusion.
Carry the reasoning to a new case
- If an interval includes zero, equality is compatible rather than proved.
- A small statistically detectable difference need not be important for the actual decision.
For the reversed interval B−A, type its lower endpoint in minutes.
Negate and reorder the endpoints: (−4.8, −1.2) becomes (1.2, 4.8).
A difference-of-means interval estimates $\mu_1 - \mu_2$. If it contains $0$, "no difference" is plausible — no convincing evidence the means differ; if it's entirely above or below $0$, there is convincing evidence (with direction). Justify the conclusion about the two population means in context.