Confidence Intervals and Estimation

Estimating with Honesty About Uncertainty

A single number from a sample (a point estimate like $\bar{x}$) almost never equals the true parameter exactly. A confidence interval does better: it gives a range of plausible values together with a stated confidence level, honestly conveying the uncertainty of the estimate.

A confidence interval for the mean has the form $$\bar{x} \pm (\text{critical value}) \times \text{standard error},$$ i.e. the point estimate plus or minus a margin of error. The margin combines a critical value (from the normal or t distribution, set by the confidence level — about $1.96$ for 95%) with the standard error $\frac{\sigma}{\sqrt{n}}$. A common 95% interval is roughly $\bar{x} \pm 2\,\text{SE}$.

Two levers control the interval's width:

• Higher confidence (99% vs 95%) uses a larger critical value, giving a wider interval — more certainty of capture requires a bigger net.
• Larger sample size $n$ shrinks the standard error, giving a narrower interval — more data means more precision.

The interpretation of "95% confidence" is subtle and the source of the most common error. It does not mean "there is a 95% probability the true parameter lies in this particular interval" — the true parameter is fixed, and this interval either contains it or not. Rather, the confidence level describes the procedure: if you repeated the sampling many times, about 95% of the intervals constructed this way would contain the true parameter. Confidence is a property of the method, not of any single interval.

Common pitfall: interpreting a 95% confidence interval as "95% probability the true value is in this interval." The parameter is fixed and the interval is what varies across samples — so a computed interval either does or does not contain the truth. The 95% refers to the long-run success rate of the procedure: about 95% of such intervals capture the parameter. It is a statement about the method, not a probability about one specific interval.

A vertical stack of accent confidence intervals from repeated samples around a fixed true-value line, about 95% of them crossing the line and a few missing — confidence as a property of the procedure.

$$\bar{x} \pm z \cdot \frac{\sigma}{\sqrt{n}}$$