Statistics Quest Part of Formula Quest

t interval for a mean (σ unknown)

With σ estimated by the sample SD, the interval for μ is x̄ ± t(n − 1)·s/√n.

\xbar \pm t_{1-\alpha/2,\;n-1}\,\frac{s}{\sqrt{n}}

Two-sided one-sample t interval

$s$: sample standard deviation (divisor $n - 1$ )
$t_{1-\alpha/2,\;n-1}$: t quantile with $n - 1$ degrees of freedom
$n$: sample size

When to use

The standard interval for a population mean from one sample. For one-sided alternatives the engine returns a one-sided bound (as R's t.test).

Assumptions

Random sample, independent observations.
Population approximately normal, or $n$ large enough. The t interval is robust to mild skewness; with small $n$ , strong skewness or outliers, consider a bootstrap interval or a median-based method.

How to interpret

Same as any confidence interval: the procedure captures μ in about $100(1-\alpha)\%$ of samples when assumptions hold.

Common mistakes

Using $z$ instead of $t$ for small samples (too narrow).
Using df $= n$ instead of $n - 1$ .
Computing it on paired data as two separate intervals; analyse the differences instead (paired interval).

Example

Data 12, 15, 11, 14, 13, 16, 12, 15: $\xbar = 13.5$ , $s = 1.7728$ , $\SE = 0.6268$ , $t_{0.975,7} = 2.3646$ . 95% interval: $(12.02, 14.98)$ .