Confidence Interval for Population Mean (σ Known)
Learning Objectives
- Calculate and interpret the confidence interval for a population mean with a known standard deviation
- Find the sample size required to estimate a population mean with a given confidence level
Why This Matters
Every polling firm targeting "±3 points at 95% confidence" runs the formula n = (z*σ/E)² before hiring a single interviewer. That equation determines whether the poll costs $40,000 or $400,000, whether it takes two weeks or two months, and whether the client gets a precise answer or a shrug. The same math sizes pharmaceutical trials, factory inspection programs, and academic research studies. Learn this formula and you understand how every quantitative study decides how much data is enough.
How to Use This Simulation
- In Compute CI mode, adjust the confidence level, sample size, and standard deviation, then watch the formula, distribution curve, and interval bounds update in real time.
- Toggle to Find Required Sample Size mode, enter a desired margin of error, and see the required n computed instantly.
- Try halving the desired margin of error and watch what happens to the required sample size -- the result will surprise you.
- Check the Explanation Panel below -- it updates as you interact and tells you why the numbers change.
The maximum acceptable margin of error for your study.
What's Happening
Quick Check
A quality control team increases their sample from n = 40 to n = 160 parts while keeping the confidence level at 95% and σ unchanged. Which of the following correctly describes what changes?
Try This
A campus health clinic sampled n = 64 students and found an average nightly sleep of x̄ = 6.8 hours. The population standard deviation is known to be σ = 1.5 hours. Compute a 95% confidence interval by hand using the formula x̄ ± z* × (σ/√n). Then enter the values into the simulation and verify your bounds match. Write one sentence interpreting this interval using the procedure-based framing: "If we repeated this sampling process many times, approximately 95% of the intervals we calculate would capture the true mean."
A city transit authority wants to estimate the average commute time with a margin of error no larger than 2 minutes at 95% confidence. From prior studies, σ = 12 minutes. (1) Compute the required sample size using n = (z*σ/E)². (2) Round appropriately and verify against the simulation. (3) Now compute the required n if the authority wants a 1-minute margin of error instead. In one sentence, explain why halving the desired margin of error did not simply double the required sample size.
A consumer advocacy group has a $30,000 budget to survey households about monthly utility bills. Each surveyed household costs $15 to contact and process. Prior data shows σ = $45. The group is considering three precision targets: (1) ME = $3 (tight), (2) ME = $5 (moderate), (3) ME = $10 (loose), all at 95% confidence. Compute the required n for each target. Determine which options fit within the budget (maximum 2,000 households at $15 each). Recommend a precision target to the group in two sentences, addressing both the statistical precision gained and the practical budget constraint. If the tightest option exceeds the budget, propose a specific tradeoff -- changing the confidence level, accepting a wider margin, or reducing σ through better measurement procedures.
Instructor Notes
Teaching Notes
This simulation is the capstone of the four-simulation confidence interval arc. Students arrive having learned the structural decomposition (Sim 19) and the correct interpretation (Sim 20). Sim 21 is where they perform the complete computation.
The most effective sequence: start in Compute CI mode with the Election Polling preset. Let students adjust n and watch ME shrink. Then ask: "If you want ME = 0.3 instead of 0.4, how much more data do you need?" Most will guess "a little more." Toggle to Find Required Sample Size, enter E = 0.4, note the n, then enter E = 0.2. The quadrupling of n is the aha moment.
Shared parameters (confidence level and σ) persist across both modes so students can see that only the direction of the calculation changes, not the underlying formula. The round-trip auto-fill (computed n from Size mode fills the n slider in CI mode) makes the structural unity visceral.
Common Student Errors
- Believing that doubling n halves ME. The square root relationship means doubling n only reduces ME by a factor of √2 ≈ 0.71 (about 29%). This is the simulation's primary misconception target.
- Confusing z* with the test statistic or believing z* changes when n changes. z* depends only on the confidence level.
- Rounding down instead of up in the sample size formula. Rounding down produces a ME slightly larger than the target, violating the precision requirement.
- Forgetting that the "σ known" assumption is a simplification. In practice, σ is rarely truly known, and the t-distribution would be used instead.
Discussion Questions
- Why does precision get more expensive as the interval gets tighter? What does this mean for a research team with a fixed budget?
- A news article says a poll has a "margin of error of ±3%." What three numbers (confidence level, σ, n) determined that margin? Could you reconstruct them?
- If a pharmaceutical company wanted to cut its trial's margin of error from 2 units to 1 unit, how would the cost change? Why might the company accept the wider margin instead?
Exam Connection
Typical exam questions give x̄, σ, n, and a confidence level and ask students to compute the CI (forward calculation). The second common format gives σ, a desired ME, and a confidence level and asks for the required n (inverse calculation). This simulation practices both. The Stretch challenge specifically targets the inverse calculation with the rounding convention, which is a frequent exam question.