Intermediate level · Module 3: Statistical Inference · Lesson 1 of 3
Confidence Intervals
Quantify the precision of estimates with standard errors and confidence intervals.
In this lesson you will
- Calculate confidence intervals for proportions and means
- Interpret interval width and overlap correctly
Key concepts
standard error95% confidence intervalinterval widthcomparing intervals
The lesson
The standard error measures how much an estimate would vary between samples. For a proportion p from n respondents it is √(p(1 − p)/n); a 95% interval is roughly the estimate ± 1.96 standard errors.
Intervals narrow as samples grow: quadrupling the sample halves the width.
Overlapping intervals do not prove that two groups are the same, and non-overlapping intervals are only a rough sign of a difference. Use a proper test to compare groups.
Step by step
- Calculate a proportion's standard error by hand.
- Build the 95% interval.
- Calculate intervals for each district.
- Compare a large and a small district.
How precise is '42% of households lack safe water'?
A 95% confidence interval for a survey proportion, worked step by step.
| Item | Working | Amount |
|---|---|---|
| Sample proportion p | 168 ÷ 400 | 0.42 |
| Standard error | √(0.42 × 0.58 ÷ 400) | 0.0247 |
| Margin of error (95%) | 1.96 × 0.0247 | 0.048 |
| Lower limit | 0.42 − 0.048 | 37.2% |
| Upper limit | 0.42 + 0.048 | 46.8% |
| 95% confidence interval | rounded | 37% to 47% |
Explanation
Plain language: our best estimate is 42% of households, and the true share in the county is very likely between 37% and 47%.
With only 100 households the margin would be about twice as wide (roughly ±10 points). This formula assumes a simple random sample; clustered or weighted surveys need survey-aware methods, which usually give wider intervals.
Check your understanding
The interval is centred on the estimate: 42% − 4.8 points and 42% + 4.8 points.
Practice activity
Report attendance indicators with 95% confidence intervals.
Expected result: Correct intervals with plain-language statements.
Check your understanding
-
How does the interval width change if the sample size is multiplied by four?
Show answer
It is roughly halved.
-
Do overlapping intervals prove two groups are the same?
Show answer
No.
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