A 95% confidence interval $(L, U)$ means:
“We are 95% confident that the true population parameter lies between $L$ and $U$.”
This does not mean there is a 95% probability that the parameter is in this particular interval — the parameter is fixed; it is either in the interval or not. The confidence level refers to the long-run reliability of the procedure.
| Factor | Effect on CI width |
|---|---|
| Larger sample $n$ | Narrower (more precise) |
| Larger variability $\sigma$ | Wider |
| Higher confidence level (e.g. 99%) | Wider |
| Lower confidence level (e.g. 90%) | Narrower |
A study of 80 Year 12 students’ weekly study hours gives a 95% CI of $(12.4, 15.8)$ hours.
Correct interpretation: “We are 95% confident that the true mean weekly study time for all Year 12 students is between 12.4 and 15.8 hours.”
Incorrect: “95% of students study between 12.4 and 15.8 hours.” (This confuses individual variability with a CI.)
If a claimed population value falls outside the 95% CI, there is evidence (at the 5% significance level) to doubt the claim.
A manufacturer claims the mean weight of their bags of flour is 1000 g. A sample gives a 95% CI of $(982 \text{ g},\; 995 \text{ g})$.
Since 1000 g is outside the interval, the data provides evidence that the true mean weight is less than 1000 g.
If the 95% CIs for two groups do not overlap, there is evidence of a significant difference between their population means.
If they overlap, we cannot conclude a significant difference from the CIs alone (a hypothesis test would be needed).
A 95% CI for a proportion of $(0.42, 0.61)$ means we are 95% confident that between 42% and 61% of the population has the characteristic of interest.
VCAA FOCUS: VCAA questions require students to write a sentence interpreting a CI in context, including the confidence level, the parameter, and the units. Generic answers without context will not receive full marks.
COMMON MISTAKE: Writing “there is a 95% probability that the mean is in this interval” after the sample has been collected. The interval is fixed; use “95% confident” instead of “95% probability”.
