Informational
How to calculate a confidence interval
For a mean, the interval is x̄ ± (critical value × SE), where SE is σ/√n if σ is known, or s/√n if you estimate it. The critical value is z* when σ is known and t* when you use s.
What the interval is for
A point estimate (the sample mean) is a single guess. A confidence interval gives a range of plausible values for the unknown population mean, at a stated confidence level such as 90%, 95%, or 99%.
Formulas
Common z* values: 1.645 (90%), 1.96 (95%), 2.576 (99%). t* is larger than z* for small df and is read from a t table.
Worked example (known σ)
n = 36, x̄ = 12.4, σ = 3.0, 95% confidence.
- SE = 3.0 / √36 = 0.5.
- Margin of error = 1.96 × 0.5 = 0.98.
- Interval: 12.4 − 0.98 to 12.4 + 0.98 → (11.42, 13.38).
We are 95% confident that the population mean lies between 11.42 and 13.38 (in the problem’s units).
Useful tips
- Larger n shrinks the interval. Higher confidence widens it.
- If the homework gives only a list of values, compute x̄ and s first — see mean and standard deviation.
- A CI and a two-sided t-test at the same α are two views of the same arithmetic.
Common mistakes
- Saying “there is a 95% chance the mean is in this interval” after the interval is computed. The long-run coverage idea belongs to the method, not this one realized interval.
- Using z* with s and n = 12.
- Forgetting to divide σ by √n (using σ as if it were already the standard error).
How Statistics Problem Solver Pro helps
The Confidence Interval calculator is free and offline. For a paragraph-style problem, scan it to see substitutions and the formula the app used. The listing also includes a t-test calculator, which is the hypothesis-test twin of a mean CI.
Check a confidence interval on iPhone
Use the free CI calculator, or scan the full homework item.
Related guides
FAQ
When do I use t instead of z for a CI?
When the population SD is unknown and you plug in the sample SD s. Degrees of freedom are n − 1 for a one-sample mean.
Learn the matching t-test steps →Is there a confidence interval app for iPhone?
Yes. Statistics Problem Solver Pro includes a CI calculator that the listing says is free and offline.
See the calculator list →