Key takeaway: A confidence interval gives a span of believable values for an unknown population quantity, found by taking a sample estimate and adding and subtracting a margin of error at a stated confidence level.
Grade 12 – AP StatisticsAP Statistics Unit 3Statistics and Data
Definition of Confidence Interval
A confidence interval turns a single estimate into a range. Instead of saying that 52% of voters favor a measure, it says the true share is likely between two limits. The width reflects the uncertainty from sampling.
The confidence level, such as 95%, describes the method: if the same procedure were used on many random samples, about 95% of the intervals produced would capture the true parameter. Any one interval either does or does not contain it.
Formula and notation
Every interval has the shape estimate plus or minus margin of error. For a proportion:
\[ \hat{p}\ \pm\ z^{*}\sqrt{\frac{\hat{p}(1-\hat{p})}{n}}. \]
The critical value is \(z^{*}=1.96\) for 95% and \(z^{*}=1.645\) for 90%. A higher level or a smaller sample gives a wider interval.
Key facts
- The confidence level refers to the long-run success rate of the method, not one interval.
- Raising the level from 90% to 99% makes the interval wider.
- A larger sample makes the interval narrower for the same confidence level.
- A confidence interval is about the population parameter, not about individual values.
Confidence Interval example
In a random sample of 600 voters, 312 support a new library levy. Build a 95% interval for the proportion of all voters who support it.
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Sample proportion: \(\hat{p}=\frac{312}{600}=0.52\).
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Sampling error per estimate: \(SE=\sqrt{\frac{0.52\cdot 0.48}{600}}\approx 0.0204\).
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Multiply by the 95% multiplier: \(1.96\times 0.0204\approx 0.040\).
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Interval: \(0.52\pm 0.04\), that is \((0.48,\ 0.56)\).
We are 95% confident that between 48% and 56% of all voters support the levy.
Common mistake
Watch out
A student writes, “The chance is 95% that the real support level sits inside 0.48 to 0.56.”
Why it is wrong
Once the confidence interval is computed, the true proportion is a fixed number that is either inside it or not, so no probability applies to that single interval. The 95% describes the reliability of the method over many samples: about 95 of every 100 such intervals would capture the truth.
Related terms and lessons
- AP Statistics · Inference for ProportionsLessonWorksheetQuiz
- AP Statistics · Inference for MeansLessonWorksheetQuiz
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