Confidence intervals
Confidence intervals
This domain tests interval estimation: constructing and interpreting intervals for means and proportions, and understanding how confidence level, sample size, and variability affect width. Items ask for a margin of error, the correct critical value, or the proper interpretation of 95% confidence.
Core rules
- General form: estimate ± critical value × standard error.
- Mean, σ known: x̄ ± z* σ/√n. Mean, σ unknown: x̄ ± t* s/√n with df = n − 1. The t critical value exceeds z and grows as df shrinks.
- Proportion: p̂ ± z* √(p̂(1−p̂)/n), valid when np̂ and n(1−p̂) are each at least about 10.
- Common critical values: 90% z = 1.645, 95% z = 1.96, 99% z = 2.576.
- Width = 2 × margin of error. Width rises with confidence level and variability, and falls as √n grows; quadrupling n halves the margin.
- Planning: n = (z*σ/E)² for a target margin E.
- A 95% interval does not mean the parameter has a 95% probability of lying inside this particular interval. It means that 95% of intervals built this way capture the parameter.
- A two-sided α-level test rejects H0 exactly when the hypothesized value falls outside the corresponding 1 − α interval.
Worked example
A sample of 25 packages has x̄ = 50 g and s = 10 g; σ is unknown. With df = 24 and 95% confidence, t* = 2.064. Standard error = 10/√25 = 2. Margin = 2.064 × 2 = 4.128. Interval: 50 ± 4.13, or (45.87, 54.13) g. Using z = 1.96 would give a slightly narrower, unjustified interval because s estimates σ.
Traps
- Saying 'there is a 95% chance the true mean is between 45.87 and 54.13'; the probability statement concerns the procedure, not this fixed interval.
- Using z when σ is unknown and n is small; use t.
- Claiming a 99% interval is more likely to contain the sample mean - it is wider, not more precise.
- Thinking a CI covers 95% of individual observations; that is a prediction or tolerance interval.
- Checking independence of two groups by whether their intervals overlap; overlapping intervals do not imply a non-significant difference.
- Forgetting to use p̂ = 0.5 for the most conservative sample-size planning when no estimate exists.