📈 Intro to Statistics · Lesson 6 of 7
Confidence intervals
Estimate ± margin of error, and what “95% confident” means.
Estimate ± margin of error
A sample statistic won’t exactly equal the population parameter. A confidence interval gives a range of plausible values: statistic ± margin of error.
Example: a proportion
A random sample of 1,000 US adults finds 58% support a policy. For a proportion, the 95% margin of error ≈ 1.96 × √(p(1 − p)/n) = 1.96 × √(0.58 × 0.42 / 1000) ≈ 0.031. The interval is 58% ± 3.1%, or 54.9% to 61.1%.
What 95% confident means
If we repeated the sampling many times and built an interval each time, about 95% of those intervals would contain the true population value. It does NOT mean there is a 95% chance the true value is in this particular interval — the true value is fixed; the interval varies.
What changes the width
| Larger sample | Narrower interval (margin shrinks like 1/√n) |
|---|---|
| Higher confidence (99%) | Wider interval |
| More variability | Wider interval |
| Bias | Not fixed by any formula — a biased sample gives a confidently wrong answer |
Tip
💡 To cut the margin of error in half, you need about four times the sample size.
✏️ Practice: your turn
- Compute a 95% interval for 40% support in a sample of 400
- Explain “95% confident” in your own words without the word “chance”
- Find a poll in the news and identify its margin of error
- Explain why a huge online poll can still be wrong
✏️ Scientific calculator with a statistics tab (mean, median, standard deviation)
🧠 Quick quiz
Answer all 4 questions correctly to mark this lesson as done.
0 of 4 correct
All lessons in Intro to Statistics
- 🎲 1. Data, variables and samplingCategorical vs. quantitative, populations, samples and bias
- 🎯 2. Mean, median and modeMeasures of center, and which one to use when data are skewed
- ↔️ 3. Spread: range, IQR and standard deviationHow spread out the data are, and how to read a box plot
- 🔔 4. The normal curve and z-scoresThe 68–95–99.7 rule and comparing values on different scales
- 📉 5. Correlation and regressionScatterplots, r, the line of best fit — and causation
- 🎯 6. Confidence intervalsEstimate ± margin of error, and what “95% confident” means
- ⚗️ 7. Hypothesis tests and p-valuesNull and alternative, the p-value, significance and errors