The Confidence Intervals
A single estimate, like a sample mean, hides its uncertainty. A confidence interval reports a range of plausible values along with how confident the method is.
What it means
A 95 percent confidence interval is built by a procedure that, across many repeated samples, would capture the true parameter about 95 percent of the time. The confidence is a property of the method, not of any one interval. For a specific computed interval the true value is either inside or not.
How width is set
An interval centers on the estimate and extends by a margin of error, which is a critical value times the standard error. Several factors shrink the width:
- A larger sample lowers the standard error, narrowing the interval.
- Less variability in the data narrows it.
- A lower confidence level narrows it but captures the truth less often.
Link to hypothesis tests
A 95 percent interval and a test at alpha 0.05 agree: if a hypothesized value falls outside the interval, the corresponding test rejects it. The interval also shows the effect size, which a bare p value hides.
Key idea
A confidence interval is a range built so the procedure captures the true parameter a set fraction of the time, narrowing with larger samples and showing effect size a p value hides.