Concept and reasoning
A parameter is a fixed but unknown population quantity, while a sample estimator is random across hypothetical samples. A confidence interval augments a point estimate by making uncertainty and method assumptions visible.
For independent normal observations with unknown σ, (X̄−μ)/(s/√n) follows t(n−1) exactly when the population is normal. Approximate procedures can be reasonable at larger n under suitable conditions; strong skewness and small samples call for caution.
A two-sided 95% t interval is X̄ ± t_(0.975;n−1) × s/√n. Under its assumptions the procedure covers the fixed μ in about 95% of repeated samples. The parameter does not move randomly inside the one observed interval.
A narrow interval does not guarantee the sample represents the population. Selection, measurement error and independence remain central to interpretation.
Formula
CI_95: X̄ ± t_(0.975, df=n−1) × s/√n
Worked example
- In a fictional sample, n=25, sample mean 68 and s=10. Standard error is 10/5=2.
- Degrees of freedom are 24 and t_(0.975;24)≈2.064. Margin is 2.064×2≈4.128.
- The resulting 95% t interval is about [63.87,72.13]. Coverage depends on model and sampling conditions.
Independent exercises
- For n=16, sample mean=50, s=8, find SE and degrees of freedom.
- With t_(0.975;15)≈2.131, compute the 95% interval.
- Explain what the 95% in a confidence interval means under its assumptions.
Written solutions stay in private instructor notes.
Base R verification
xbar <- 68; s <- 10; n <- 25
tcrit <- qt(0.975, df=n-1)
se <- s/sqrt(n)
interval <- xbar + c(-1,1)*tcrit*se
c(tcrit=tcrit, se=se, lower=interval[1], upper=interval[2])