Confidence Intervals: Capturing the True Mean

What does "95% confidence" actually mean?

A 95% confidence interval means: if you repeated the experiment many times, about 95% of the computed intervals would contain the true mean. It does NOT mean "there's a 95% chance the true mean is in this particular interval" — the true mean is fixed; the intervals are random.

The formula

$\text{CI} = \bar{x} \pm z^* \cdot \frac{s}{\sqrt{n}}$

where $\bar{x}$ is the sample mean, $s$ is the sample standard deviation, $n$ is the sample size, and $z^* = 1.96$ for 95% confidence.

Step-by-step example

Given: A sample of $n = 30$ students has $\bar{x} = 75$, $s = 10$.

Step 1 — Compute the standard error: $SE = s/\sqrt{n} = 10/\sqrt{30} = 1.826$.

Step 2 — Compute the margin of error: $ME = 1.96 \times 1.826 = 3.58$.

Step 3 — Build the interval: $75 \pm 3.58 = (71.42, 78.58)$.

Interpretation: We are 95% confident that the true population mean is between 71.42 and 78.58.

The simulation insight

Take 50 random samples and compute a 95% CI from each. About 47-48 will contain the true mean; 2-3 will miss. This is the "95%" in action — not every interval captures the truth, but 95% do over the long run.

Try it in the visualization

50 horizontal CI bars are drawn. About 95% (green) capture the true mean line; a few (red) miss it. Adjust the confidence level and sample size to see how the intervals change.

$\text{CI} = \bar{x} \pm z^{*} \cdot \dfrac{s}{\sqrt{n}}$, where $z^{*} = 1.96$ for 95%.