The idea
A sample mean is a single guess for the population mean. A confidence interval adds an honest margin for sampling error:
estimate ± margin of error
The margin of error = critical value × standard error.
Interpreting the confidence level
A 95 percent confidence level means: if we repeated the sampling process many times and built an interval each time, about 95 percent of those intervals would capture the true parameter. After a particular interval is computed, the parameter is either inside it or not; the 95 percent describes the method, not a probability for this one interval.
Interval for a mean
When σ is known (or n is large):
x̄ ± z* · σ/√n
Common critical values: z* = 1.645 for 90 percent, 1.96 for 95 percent, 2.576 for 99 percent.
When σ is unknown and estimated by s, use the t distribution with n − 1 degrees of freedom: x̄ ± t* · s/√n. The t curve has heavier tails, which matters most for small samples.
Interval for a proportion
p̂ ± z* · √(p̂(1 − p̂)/n)
What changes the width
- Higher confidence level → larger critical value → wider interval. You pay for more certainty with less precision.
- Larger sample size → smaller SE → narrower interval.
- More variable data (bigger σ) → wider interval.
Worked example 1: a mean
A sample of n = 36 packets has a mean mass of x̄ = 50 g. Assume σ = 12 g. Build a 95 percent CI.
- SE = 12 / √36 = 12 / 6 = 2.
- Margin = 1.96 × 2 = 3.92.
- Interval = 50 ± 3.92 = (46.08, 53.92).
We are 95 percent confident that the true mean mass is between about 46.1 and 53.9 g.
Worked example 2: a proportion
In a poll of n = 100 people, 40 prefer option A, so p̂ = 0.40.
- SE = √(0.4 × 0.6 / 100) = √0.0024 ≈ 0.049.
- Margin = 1.96 × 0.049 ≈ 0.096.
- Interval ≈ (0.304, 0.496), or about 30 to 50 percent.
With n = 400 instead, SE halves to about 0.0245 and the margin shrinks to about 0.048.
Reading an interval
Given an interval (12, 18), the point estimate is the midpoint, 15, and the margin of error is half the width, 3. For an interval (4.2, 5.8), the margin is (5.8 − 4.2)/2 = 0.8.
If a claimed value lies outside the interval, the data are not consistent with that value at that confidence level.
Common mistakes
- Saying there is a 95 percent chance the parameter is in this specific interval.
- Using z when the sample is small and σ is unknown.
- Forgetting that the interval describes the population mean, not individual values.