Module 06 · Confidence intervals
Confidence intervals with z.
Build a z confidence interval for a population mean when the population standard deviation is known. See how precision changes with the sample size.
See the interval take shape
Known population σInterval-construction illustration: a normal reference curve centered at x̄, with width set by the standard error. This is not a probability distribution for the unknown population mean μ.
The 95 percent z confidence interval is 46.08 to 53.92, with standard error 2 and critical value 1.960.
The idea
An interval pairs the sample mean with a margin of error: z* × σ / √n. More data reduce the standard error. A higher confidence level uses a larger critical value and gives a wider interval.
At 95% confidence, about 95% of intervals constructed this way over repeated independent samples would contain the fixed population mean.
TRY THIS Four times the data
Keep the confidence level and σ fixed. Increase n from 25 to 100: the standard error and the margin of error are each cut in half.
Assumptions, notation & reference
Use an independent random sample with a known population standard deviation σ. This interval has exact coverage for a normal population; otherwise, its accuracy depends on a suitable normal approximation for the sample mean. Small samples from strongly skewed populations may not justify that approximation.
Let C be the confidence level as a proportion and α = 1 − C. The positive critical value is z* = Φ⁻¹(1 − α/2). Each unshaded tail has area α/2. Tail labels refer to the entire tails, including areas beyond the displayed range. The plot height is normalized for comparison; the reported limits are on the original measurement scale.
A particular computed interval either contains μ or does not; 95% describes the long-run coverage of the procedure. When σ is estimated from the sample, explore the t interval.
Reference: NIST / SEMATECH e-Handbook: Confidence intervals ↗