One-sample t-test

Tests whether a population mean equals a stated value, using the sample mean, the sample SD and the t distribution with n − 1 degrees of freedom.

t = \frac{\xbar - \mu_0}{s/\sqrt{n}}, \qquad df = n - 1

$\xbar$: Sample mean.
$\mu_0$: Mean stated by $H_0$ .
$s$: Sample standard deviation (divisor n − 1).
$n$: Sample size.

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

The matching confidence interval (one-sided alternatives give a one-sided bound).

$t_{1-\alpha/2,\,n-1}$: The t quantile with n − 1 df.

When to use

One quantitative variable, one sample, and a benchmark mean to compare with (a target, a specification, a claimed value). If σ were genuinely known, the z-test would use $\sigma$ instead of $s$ — in practice σ is almost never known.

Assumptions

Independent observations (random sample or a process in a stable state).
The population is roughly normal, or n is large enough for the CLT: with n ≥ 30 moderate skewness is fine; with small n check a dot plot or Q–Q plot for strong skew and outliers.
Outliers matter more than mild non-normality: the mean and SD are both sensitive to them.

How to interpret

A small p-value is evidence that the population mean differs from $\mu_0$ (in the direction of $H_1$ ).
The CI gives the plausible values of μ; the test at level α rejects exactly when $\mu_0$ is outside the $(1-\alpha)$ CI.
Cohen's $d = (\xbar - \mu_0)/s$ expresses the gap in SD units.

Common mistakes

Using the population SD formula (divide by n) for s.
Using z critical values with a small sample and estimated SD.
Testing paired before/after data as two separate samples — use the paired t-test on the differences.
Concluding "the mean is 12" after a non-significant result.

Example

Delivery times 12, 15, 11, 14, 13, 16, 12, 15 (n = 8) vs $\mu_0 = 12$ : $\xbar = 13.5$ , $s = 1.773$ , SE $= 1.773/\sqrt 8 = 0.627$ , $t = 1.5/0.627 = 2.39$ , df = 7, two-sided $p = 0.048$ . 95% CI: 12.02 to 14.98 minutes; $d = 0.85$ .