Confidence Intervals

A confidence interval reports a range of plausible values for a parameter—a mean incubation time, a prevalence—together with an estimate’s uncertainty. It shifts attention from a single point estimate to the precision behind it.

Across repeated samples, about 95% of 95% confidence intervals cover the true mean (misses highlighted).

The idea

A confidence interval (CI) is an interval, computed from data, designed so that the procedure captures the true parameter a specified fraction of the time (the confidence level, e.g. 95%) across repeated samples.

For a population mean with unknown variance, the standard $100(1-\alpha)\%$ CI is

$\bar{x} \pm t_{\alpha/2,\,n-1}\,\frac{s}{\sqrt{n}},$

where $\bar{x}$ is the sample mean, $s$ the sample standard deviation, $n$ the sample size, and $t_{\alpha/2,\,n-1}$ the upper- $\alpha/2$ critical value of the $t$ -distribution with $n-1$ degrees of freedom. The term $s/\sqrt{n}$ is the standard error.

Correct interpretation

The frequentist statement is about the procedure, not a fixed interval:

If we repeated the sampling many times and built a 95% CI each time, about 95% of those intervals would contain the true parameter.

For one particular observed interval, the parameter either is or is not inside it—so it is incorrect to say $\Pr(\theta \in \text{CI}) = 0.95$ for that fixed interval. The 95% describes long-run coverage of the method (see sampling distributions).

Worked example

For $n=10$ , $\bar{x}=5.8$ , $s=1.2$ , a 95% CI for the mean uses $t_{0.025,9}\approx 2.262$ :

$5.8 \pm 2.262 \times \frac{1.2}{\sqrt{10}} = 5.8 \pm 2.262 \times 0.3795 = 5.8 \pm 0.858,$

giving $(4.94,\ 6.66)$ . Because the reference value $5$ lies inside, this agrees with a two-sided $t$ -test at $\alpha=0.05$ that does not reject $H_0:\mu=5$ .

In code

R

set.seed(42)
x <- rnorm(10, mean = 5.8, sd = 1.2)
t.test(x, mu = 5)$conf.int      # 95% CI for the mean
$```

### Python

```python
import numpy as np
from scipy import stats
rng = np.random.default_rng(42)
x = rng.normal(5.8, 1.2, size=10)
print(stats.t.interval(0.95, df=len(x)-1,
                       loc=x.mean(), scale=stats.sem(x)))

(np.float64(4.595976442189666), np.float64(6.198652417636242))

Julia

using HypothesisTests, Random, Distributions
Random.seed!(42)
x = rand(Normal(5.8, 1.2), 10)
println(confint(OneSampleTTest(x)))   # 95% CI

Simulation: coverage

Build many 95% CIs from a known population and count how many cover the true mean—about 95% should.

set.seed(1)
mu <- 5
covered <- replicate(10000, {
  x  <- rnorm(20, mean = mu, sd = 2)
  ci <- t.test(x)$conf.int
$  ci[1] <= mu && mu <= ci[2]
})
mean(covered)   # ~0.95

Why it matters for statistics

Confidence intervals communicate both the estimate and its uncertainty on the natural scale of the parameter, making results easier to judge than a bare p-value. Their coverage guarantee is the interval-based counterpart to hypothesis testing and is central to reporting effect sizes in epidemiology.