Statistics 101 · Worked example
Confidence intervals explained
A confidence interval estimates a population value using sample data. It shows uncertainty around your estimate, rather than giving a single number.
The method
For a mean with a known population standard deviation, use x̄ ± z × σ / √n. For 95% confidence, z is approximately 1.96. If the population standard deviation is unknown, use the sample standard deviation and an appropriate t critical value instead.
Worked example
A delivery business samples 100 deliveries. The sample mean delivery time is 40 minutes, and the known population standard deviation is 10 minutes. Assume independent observations and an appropriate sampling method.
- Standard error = 10 / √100 = 1 minute.
- Margin of error = 1.96 × 1 = 1.96 minutes.
- 95% confidence interval = 40 ± 1.96 = [38.04, 41.96] minutes.
Interpret the answer
If we repeated the sampling process many times, about 95% of intervals constructed this way would contain the true population mean. For this sample, the interval estimates the mean delivery time between 38.04 and 41.96 minutes.
A common mistake
The interval does not describe where 95% of individual deliveries fall. It estimates the population mean.
Check your understanding
What happens to the margin of error when the sample size increases from 100 to 400, with everything else held constant?
Show the answer
It halves: √400 is twice √100, so the standard error and margin of error are half as large.
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