StatisticsIntroductory StatisticsHigh school

Confidence Intervals

Estimate a mean or proportion with a margin of error.

A confidence interval estimates an unknown population value from a sample and reports how uncertain the estimate is. It has the form estimate $\pm$ margin of error, for example $\bar{x} \pm z^* \frac{\sigma}{\sqrt{n}}$ for a mean. A $95\%$ confidence level means the method captures the true value in $95\%$ of repeated samples.

The margin of error shrinks as the sample size grows, but only with the square root of $n$ : quadrupling the sample halves the margin. Higher confidence levels need a larger critical value and give wider intervals, so there is always a trade-off between precision and confidence.

Key terms

Interval for a mean: Point estimate plus or minus critical value times standard error.
$\bar{x} \pm t^*\frac{s}{\sqrt{n}}$

Key ideas

Critical values: $z^* = 1.645$ (90%), $1.96$ (95%), $2.576$ (99%).
Use $t^*$ with $n - 1$ degrees of freedom when $\sigma$ is estimated by $s$ .
Proportion: $\hat{p} \pm z^*\sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}$ .
Check conditions: random sample, independence, roughly normal sampling distribution.

Worked example

A random sample of 100 bulbs has mean life 1200 hours; assume $\sigma = 50$ . Build a $95\%$ interval.

Standard error: $\frac{50}{\sqrt{100}} = 5$ .
Margin: $1.96 \times 5 = 9.8$ .
Interval: $1200 \pm 9.8$ .
$(1190.2, 1209.8)$ hours.

Answer: $(1190.2, 1209.8)$

Common mistakes

Saying there is a $95\%$ probability the true mean is in this particular interval.
Using $n$ instead of $\sqrt{n}$ in the standard error.
Ignoring the conditions for inference.

Practice questions

Try each one, then open the answer. Numerical answers on this page are verified by an automated calculation check.

1.What is the standard error when $\sigma = 12$ and $n = 36$ ?
Show answer
$2$
2.What is the $95\%$ margin of error when the standard error is 3?
Show answer
$5.88$
3.For $\hat{p} = 0.5$ and $n = 400$ , what is the standard error?
Show answer
$0.025$

Key terms

Key ideas

Worked example

Common mistakes

Practice questions

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