AP STATISTICS · INFERENCE

Confidence Intervals

Estimate a population parameter with a range of plausible values.

16 min read 3 practice questions

A sample statistic rarely equals the true parameter exactly. A confidence interval gives a range of plausible values: point estimate ± margin of error.

\hat{p} \pm z^*\sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}

One-proportion z-interval

\bar{x} \pm t^*\frac{s}{\sqrt{n}}

One-sample t-interval for a mean

Common critical values:

z^* = 1.645

for

90\%

1.96

for

95\%

, and

2.576

for

99\%

. Higher confidence means a wider interval; a larger sample size means a narrower one.

Worked example · Building an interval

In a random sample of

400

students,

240

prefer online homework. Build a

95\%

CI for the true proportion.

1
$\hat{p} = \frac{240}{400} = 0.60$ .
2
$SE = \sqrt{\frac{0.6(0.4)}{400}} = 0.0245$ .
3
$ME = 1.96(0.0245) \approx 0.048$ .
4
Interval: $(0.552,\ 0.648)$ .

CHECK YOUR UNDERSTANDING

Practice

Answer each question, then check.

Holding everything else constant, increasing the confidence level from 95% to 99% will make the interval…

A 95% CI is

(0.42, 0.58)

. What is the margin of error?

To cut the margin of error in half, you should multiply the sample size by…

Numeric answers accept decimals or fractions like 3/4.