Confidence interval with known σ (z)
With the population standard deviation known, use z; divide by √n, not n.
Worked example
A sample of n = 49 has mean x̄ = 87.3; σ = 12 is known. Find the 99% confidence interval for μ (z = 2.576).
Worked solution
- 1.σ / √n = 12 / 7 = 1.7143
- 2.Margin: z · σ / √n = 2.576 · 1.7143 = 4.4160
- 3.[87.3 − 4.4160; 87.3 + 4.4160] = [82.88; 91.72]
Typical mistake: Dividing by n instead of √n.
Your turn
Three problems of exactly this type, each time with new numbers. Comma or point; fractions like 1/6 work too.
Task 1 of 3Confidence interval
Task
A sample of n = 49 has mean x̄ = 53.1; σ = 15 is known. Find the 90% confidence interval for μ (z = 1.645).
Lower limit (2 decimal places)
Upper limit (2 decimal places)
Comma or point; fractions like 1/6 work too.
Formulas for this topic
- σ known: x̄ ± z · σ / √n
- σ unknown: x̄ ± t(n − 1) · s / √n
- Minimum n for margin E: n ≥ (z · σ / E)², round up
Typical mistake: Dividing by n instead of √n.
Frequently asked questions
What is the typical mistake in "Confidence interval with known σ (z)"?
Dividing by n instead of √n.
When do I use t instead of z?
When σ is unknown and estimated by s from the sample, especially for small samples.