Confidence Intervals

Summary

A confidence interval gives a range of plausible values for a population parameter based on sample data. Before sampling, the endpoints are random and the interval has a known probability of covering the true parameter. After sampling, the realized interval either contains the parameter or it does not.^[1]

Prerequisites

Definition / Notation

A $100 (1 - α) %$ confidence interval for a parameter $θ$ is a random interval $(L (X), U (X))$ such that

P (L (X) \leq θ \leq U (X)) = 1 - α .

Symbol	Meaning
$1 - α$	Confidence level
$z_{α / 2}$	Upper $α / 2$ quantile of $N (0, 1)$
$t_{α / 2, ν}$	Upper $α / 2$ quantile of $t_{ν}$
$χ_{α / 2, ν}^{2}$	Upper $α / 2$ quantile of $χ_{ν}^{2}$
$s$	Sample standard deviation

Parameters / Assumptions

Random sample $X_{1}, \dots, X_{n}$ .
For the mean with known $σ$ : normal population or $n$ large enough for the Central Limit Theorem.
For the mean with unknown $σ$ : approximately normal population or $n$ large.
For a proportion: $n \hat{p}$ and $n (1 - \hat{p})$ are large enough for the normal approximation.
For a variance ratio: independent normal samples.

Essential Result

Parameter	Assumptions	Confidence Interval
Mean $μ$	$σ$ known	$\bar{x} \pm z_{α / 2} \frac{σ}{\sqrt{n}}$
Mean $μ$	$σ$ unknown	$\bar{x} \pm t_{α / 2, n - 1} \frac{s}{\sqrt{n}}$
Proportion $p$	Large sample	$\hat{p} \pm z_{α / 2} \sqrt{\frac{\hat{p} (1 - \hat{p})}{n}}$
Variance ratio $σ_{1}^{2} / σ_{2}^{2}$	Independent normal samples	$(\frac{s_{1}^{2}}{s_{2}^{2}} \frac{1}{F_{α / 2, n_{1} - 1, n_{2} - 1}}, \frac{s_{1}^{2}}{s_{2}^{2}} F_{α / 2, n_{2} - 1, n_{1} - 1})$

The correct interpretation is that $100 (1 - α) %$ of intervals constructed by this procedure will cover $θ$ .^[2]

Worked Example

A sample of size $n = 25$ has mean $\bar{x} = 50$ and standard deviation $s = 10$ . A 95% confidence interval for $μ$ uses $t_{0.025, 24} \approx 2.064$ :

50 \pm 2.064 \frac{10}{\sqrt{25}} = 50 \pm 4.128 .

The interval is $(45.87, 54.13)$ .

Common Mistakes

Saying "the probability that $θ$ is in this interval is 0.95" after the data are observed.
Using the normal quantile when $σ$ is unknown and $n$ is small.
Forgetting the $\sqrt{n}$ in the standard error.
Treating a confidence interval as a prediction interval for future observations.

Connections

References

OpenStax, Introductory Statistics, "Confidence Intervals", https://openstax.org/details/books/introductory-statistics ↩︎
NIST/SEMATECH, e-Handbook of Statistical Methods, "Confidence Limits", https://www.itl.nist.gov/div898/handbook/ ↩︎