Confidence Intervals for a Mean

Definition The normal sampling model for one mean says that the observations $X_1, \dots, X_n$ are independent and identically distributed $N(\mu, \sigma^2)$, with $\mu$ and $\sigma > 0$ both unknown. Under this model the sample mean has the distribution $\bar{X} \sim N(\mu, \sigma^2/n)$. For $n \ge 2$, write $S$ for the sample SD with the $n - 1$ denominator, viewed as a random quantity. Then the standardized mean with an estimated standard error has the t distribution with $n - 1$ degrees of freedom,

for every value of $\mu$ and $\sigma$.

Definition For a confidence level $1 - \alpha$ and $\nu$ degrees of freedom, the t critical value $t_{1 - \alpha/2, \nu}$ is the $1 - \alpha/2$ quantile of the t distribution with $\nu$ degrees of freedom. If $T \sim t_{\nu}$, then $P(-t_{1 - \alpha/2, \nu} \le T \le t_{1 - \alpha/2, \nu}) = 1 - \alpha$. At a fixed level the t critical value is larger than the normal critical value $z_{1 - \alpha/2}$ and decreases toward it as $\nu$ grows.

For comparison, $z_{0.975} \approx 1.9600$.

Definition Under the normal sampling model with $n \ge 2$, the one-sample t interval for $\mu$ at confidence level $1 - \alpha$ is

The interval contains $\mu$ exactly when $-t_{1 - \alpha/2, n-1} \le T \le t_{1 - \alpha/2, n-1}$, so its coverage is exactly $1 - \alpha$ for every $\mu$ and every $\sigma$. The margin of error is $t_{1 - \alpha/2, n-1} \, s/\sqrt{n}$. When the observations are independent but the population is not normal, the coverage is generally not exactly $1 - \alpha$, and how close it comes depends on the population and the sample size.

Definition A departure from normality is any way the population distribution differs from a normal one, such as skewness, heavy tails, outliers, gaps or several peaks. If the observations are independent and identically distributed with finite positive variance but not normal, the coverage of the t interval is generally not exactly $1 - \alpha$. By the central limit theorem, and because $s$ is close to $\sigma$ with high probability in large samples, its coverage approaches $1 - \alpha$ as $n$ grows, but how large $n$ must be depends on the population, and no single sample size is enough for every population. Strong skewness, heavy tails, extreme outliers and very small samples call for the most care. Dependence between observations is a separate failure of the model, and it can make $s/\sqrt{n}$ the wrong standard error at any sample size.