Confidence Interval
In short: Confidence interval gives a band of believable values for an unknown population parameter, centered on the sample estimate, with a stated level of confidence in the recipe that produced it.
AP StatisticsStatisticsAP Statistics Unit 3
What is Confidence Interval?
A confidence interval attaches honesty to an estimate. Rather than announcing that “52% of voters approve,” it reports “between 48% and 56%,” and it names how dependable the recipe is.
Every such interval has the pattern: estimate \(\pm\) (critical value)(standard error). For a proportion that becomes \(\hat{p} \pm z^*\sqrt{\frac{\hat{p}(1-\hat{p})}{n}}\), with \(z^* = 1.96\) at 95% confidence. The level describes the method, not one interval: sample again and again, and about 95% of the bands built this way will trap the true parameter.
How to spot it
- The problem says “estimate” a mean or proportion and asks for a range of values.
- A news report lists a result with a plus-or-minus figure and a confidence level.
- You are asked to interpret what 95% confidence means, or to decide whether a claimed value is plausible.
- A question asks how the width changes with sample size or confidence level.
The method
- Verify the conditions: random sampling, independence, and enough successes and failures (or a large sample for a mean).
- Compute the point estimate and its standard error.
- Multiply the standard error by the critical value (\(z^*\) or \(t^*\)) to obtain the margin of error.
- Write the lower and upper ends, then explain them in context, naming the parameter and the confidence level.
Confidence Interval: solved example
Problem
A coffee roaster in Phoenix surveys a random sample of 400 subscribers, and 148 say they prefer dark roast. Build a 95% interval for the share of all subscribers who prefer it.
Solution
\(\hat{p} = \frac{148}{400} = 0.37\). Both \(148\) and \(252\) exceed 10, so the normal model is reasonable.
\(SE = \sqrt{\frac{0.37 \times 0.63}{400}} \approx 0.0241\), and the margin of error is \(1.96 \times 0.0241 \approx 0.0473\).
Interval: \(0.37 \pm 0.047\), that is (0.323, 0.417). The roaster can be 95% confident that between 32.3% and 41.7% of all subscribers prefer dark roast.
Try it
Question 1
What does “95% confidence” mean?
Show the answer
The method captures the true parameter in about 95% of repeated random samples. It does not mean there is a 95% chance that one particular computed interval contains it.
Question 2
How can you make a confidence interval narrower?
Show the answer
Increase the sample size, or accept a lower confidence level (for example 90% instead of 95%).
Confidence Interval vs. Margin of Error
The margin of error is the half-width: the amount added and subtracted. The interval is the complete range from the lower endpoint to the upper endpoint. In the roaster example, 0.047 is the margin of error and (0.323, 0.417) is the interval.
Keep going
Lessons
Worksheets and quizzes
- Confidence Intervals for a Proportion – Worksheet
- Confidence Intervals for a Proportion – Quiz
- t Intervals and t Tests for a Mean – Worksheet
- t Intervals and t Tests for a Mean – Quiz
Related words
Whole dictionary: A–Z · AP Statistics · Statistics