- What does a 95% confidence interval actually mean?
- If you repeated the same sampling procedure many times and built a 95% CI each time, about 95% of those intervals would contain the true population parameter. It's a statement about the procedure, not a probability that the true value is in this one interval — that subtle distinction is what trips up most stats students. In practice, the interval gives you a credible range for where the truth is likely to live, assuming your sample is representative.
- When should I use z and when should I use t?
- Use the z (standard normal) critical value when the population standard deviation σ is known, OR when the sample size is large (n ≥ 30) — by the central limit theorem the sampling distribution is close enough to normal. Use t (Student's t with n−1 degrees of freedom) when σ is unknown and n < 30. The Mean tab auto-picks for you and exposes an override if your textbook insists on a specific choice.
- Wald vs. Wilson — which proportion interval should I use?
- Wald is the textbook formula (p̂ ± z·√(p̂(1−p̂)/n)) and works well when n is large and p̂ isn't near 0 or 1. Wilson is the score interval; it stays inside [0, 1] and gives more accurate coverage near the edges. If you have x = 0 or x = n, Wald collapses to a zero-width interval — that's the signal to switch to Wilson. For most homework problems with healthy counts, Wald is fine.
- What is Welch's t-interval and why is it the default?
- Welch's interval is the two-sample t-interval that does NOT assume the two groups share the same variance. It uses the Welch–Satterthwaite formula to compute an effective degrees of freedom from the two sample variances. It's the default because the equal-variance assumption is rarely true in real data and Welch is essentially as powerful when variances actually are equal. Use Pooled only if you have a strong reason to believe both groups have the same spread.
- Why does the interpretation sentence say 'true population mean' instead of 'true mean'?
- Because that's the parameter being estimated. The sample mean (x̄) is a known number — no interval needed. The CI is a range for the unknown population mean (μ), the value you'd get if you could measure the entire population. Same logic for proportion (p̂ estimates the population proportion p) and two-sample (the observed difference x̄₁ − x̄₂ estimates the population difference μ₁ − μ₂).
- My sample has standard deviation exactly 0 — what do I do?
- Every observation in your sample is identical, so there's no variability to project from. The calculator returns a degenerate interval — lower = upper = the sample mean — and a note explaining what happened. You can't actually estimate uncertainty from a constant sample; you'd need at least one different value to get a non-zero standard error.
- Why does my answer differ slightly from SPSS or R?
- Two reasons: rounding (different tools round the critical value at different decimals, then propagate) and the t-distribution numerical approximation (different implementations of the inverse t use different algorithms with different precision). Microapp computes critical values to about six significant figures and rounds the displayed output to four. The mismatch should be in the last displayed digit at most. For published research, use the same software end-to-end and cite it.
- How big a sample do I need for a margin of error of X?
- Solve the margin formula for n. For a mean with known σ: n = (z·σ / E)². For a proportion: n = z²·p̂(1−p̂) / E², or n = z²·0.25/E² (the conservative case when you don't know p̂ in advance). For 95% CL and a margin of error 0.05, that's n ≈ 384 for an unknown proportion. The reverse-direction tool is on the roadmap; for now, compute by hand using these formulas.
