Confidence Intervals

What is a confidence interval?

A confidence interval gives a range of plausible values for the population mean, based on sample data.

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

For 95% confidence: $z^* = 1.96$.

Step-by-step

Given: $\bar{x} = 72$, $\sigma = 8$, $n = 50$.

Step 1 — Standard error: $SE = \sigma / \sqrt{n} = 8 / \sqrt{50} = 8 / 7.071 = 1.131$

Step 2 — Margin of error: $ME = 1.96 \times 1.131 = 2.217$

Step 3 — Confidence interval: $72 \pm 2.217 = (69.78, 74.22)$

$\boxed{95\% \text{ CI}: (69.78, 74.22)}$

Interpretation: We are 95% confident that the true population mean lies between 69.78 and 74.22.

What "95% confident" means

If we repeated this sampling process many times, about 95% of the computed intervals would contain the true mean. It does NOT mean there's a 95% probability the true mean is in this specific interval.

Wider vs narrower intervals

• Higher confidence (99% vs 95%) → wider interval
• Larger sample size → narrower interval
• Larger $\sigma$ → wider interval

Try it in the visualization

The bell curve shows the sampling distribution. The CI is drawn as a horizontal bar. Adjust confidence level and sample size to see the interval widen or narrow.