What is a Confidence Interval?
A confidence interval (CI) is a range of values that likely contains the true population parameter. Instead of giving a single point estimate, a CI acknowledges uncertainty by providing a range.
"We are 95% confident the true mean falls between 48.2 and 51.8"
95% CI: [48.2, 51.8]
The Formula
The confidence interval for a population mean is:
Confidence Interval Formula
CI = x̄ ± z* × (σ / √n)
- x̄ = sample mean
- z* = critical value (1.96 for 95% CI)
- σ = standard deviation
- n = sample size
- σ/√n = standard error
| Confidence Level | z* Value |
|---|---|
| 90% | 1.645 |
| 95% | 1.960 |
| 99% | 2.576 |
Correct Interpretation
Common Misconception
A 95% CI does NOT mean "there's a 95% probability the true mean is in this interval." The true mean either is or isn't in the interval—it's fixed.
Correct Interpretation
"If we repeated this sampling process many times, 95% of the calculated intervals would contain the true population mean."
Worked Examples
Example: Customer Satisfaction
You survey 100 customers and find a mean satisfaction score of 7.5 with standard deviation of 1.5. Calculate the 95% CI.
1
Find the standard error
SE = 1.5 / √100 = 0.15
2
Calculate margin of error
ME = 1.96 × 0.15 = 0.294
3
Build the interval
CI = 7.5 ± 0.294 = [7.21, 7.79]
Interpretation: We are 95% confident the true mean customer satisfaction is between 7.21 and 7.79.
What Affects CI Width?
Sample Size (n)
Larger n = narrower CI
More data = more precision
Standard Deviation (σ)
Larger σ = wider CI
More variability = less certainty
Confidence Level
Higher confidence = wider CI
99% CI is wider than 95% CI