All courses › Statistics and Risk Analysis › Confidence interval

Confidence interval

A confidence interval gives a range of plausible values for an unknown quantity, such as the mean $\mu$ , based on a sample. The standard error of the mean shrinks with $\sqrt n$ , so a larger sample gives a narrower interval. A higher confidence level gives a wider interval.

\bar x \pm z_{\alpha/2}\,\frac{\sigma}{\sqrt n}

confidence interval for

\mu

with known

\sigma

SE = \frac{\sigma}{\sqrt n}

standard error of the mean

Symbols

$\bar x$	sample mean
$z_{\alpha/2}$	critical z-value, 1.96 for 95%
$n$	sample size

Example

$\bar x = 35$ , $\sigma = 18$ , $n = 36$ : $SE = 18/6 = 3$ .

Upper bound of 95% interval: $35 + 1.96\cdot 3 \approx 40.88$ .

A wider interval means higher confidence, not worse data — confidence level and precision are traded off.

Practise inference and risk for free →

← Hypothesis testing and p-value · Type I and type II errors →

Part of Statistics and Risk Analysis: Inference and risk.