Question 1

What does a 95% confidence interval mean?

Accepted Answer

A 95% CI does NOT mean there is a 95% probability the true parameter lies in this interval — the parameter is fixed. The correct interpretation: if you collected many samples and built a 95% CI from each, approximately 95% of those intervals would contain the true population value. For a single interval, you can say: "I used a procedure that captures the true value 95% of the time in repeated sampling."

Question 2

When should I use a t-interval vs a z-interval?

Accepted Answer

Use t-interval for the population mean when σ is unknown (the usual case) and either the population is normal or n ≥ 30. Use z-interval when σ is known (rare in practice) or n is very large (n > 200) where t and z converge. For a proportion CI, always use z when np̂ ≥ 5 and n(1−p̂) ≥ 5 (normal approximation condition). For small proportions with small n, the Wilson score interval is more accurate.

Question 3

How does sample size affect the confidence interval?

Accepted Answer

Larger samples produce narrower (more precise) intervals. Margin of error scales as 1/√n — quadrupling the sample size halves the margin of error. Example: n=100, s=10, 95% CI → MOE ≈ ±2.0; n=400 → ±1.0; n=1600 → ±0.5. Each doubling of precision costs 4× as many observations. Use the sample size calculator to determine n for a target margin of error.

Question 4

What is the margin of error?

Accepted Answer

MOE = half the CI width = t* × SE or z* × SE. It is the maximum expected distance between the sample estimate and the true population value at your chosen confidence level. For political polls: "Candidate A leads 52% ± 3%" means the 95% CI is [49%, 55%]. If the CI includes 50%, the lead is not statistically significant at that confidence level.

Question 5

Can the confidence interval include negative values?

Accepted Answer

Yes, if the true population mean could plausibly be negative. Example: a CI for mean temperature difference [−0.5°C, 2.3°C] includes zero — the difference is not statistically significant. For proportions, the CI should be bounded to [0%, 100%]. The simple normal approximation CI can produce values outside [0,1] for small samples — use the Wilson score interval instead.

Question 6

What is the relationship between confidence intervals and hypothesis testing?

Accepted Answer

A 95% CI and a two-tailed α=0.05 hypothesis test are mathematically equivalent: if the hypothesised value μ₀ falls outside the 95% CI, the test rejects H₀ at α=0.05 (and vice versa). CIs are often preferred over p-values because they communicate both statistical significance and the magnitude and precision of the estimate — giving a richer picture than a single binary decision.

Situation	Method	Formula	When it fails
Mean, σ unknown (usual)	t-interval	x̄ ± t* × s/√n	n < 30 and non-normal data
Mean, σ known (rare)	z-interval	x̄ ± z* × σ/√n	σ almost never known in practice
Proportion, large n	Normal approx (z)	p̂ ± z* × √(p̂(1-p̂)/n)	np̂ < 5 or n(1−p̂) < 5
Proportion, small n or p near 0/1	Wilson score	Complex — see R prop.test()	Simple formula breaks near 0 or 1
Large n (n > 200)	z or t (same result)	Either — t → z for large df	No failure — results identical
Difference of two means	Two-sample t or Welch	Welch t preferred (unequal σ)	Equal-variance t wrong when σ differ

Confidence Interval Calculator

Formula

Choosing Your CI Method — z vs t vs Wilson Score

Case Study: Political Poll Margin of Error

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