Statistical inference & estimation
The idea
Inference runs probability in reverse: instead of deducing data from a known distribution, you observe a sample and reason back to the population that produced it. The pivotal fact is the central limit theorem — for large samples, the sample mean is approximately normal around the true mean μ with standard error σ/√n. Because the standard error shrinks like 1 over the square root of n, averaging is a noise-suppression machine, and you can say precisely how much trust a given sample size buys.
A confidence interval converts that into a usable statement: sample mean ± (critical value) × (standard error), where the 95% critical value of 1.96 comes from the normal distribution. The procedure traps the true mean in 95% of repetitions — confidence describes the long-run reliability of the method, not a probability about any single computed interval.
That is also the misconception to root out: once an interval like (500.0, 504.0) is computed, the true mean is not 95% likely to be inside it — μ is a fixed number, in or out, and randomness lived in the sampling, not in μ. Saying 'the method captures the truth 19 times in 20' is both correct and more honest.
Worked example
A coffee roaster's filling machine has a known standard deviation of σ = 8 g. A quality check of n = 64 randomly chosen bags has a sample mean of 502 g. Build a 95% confidence interval for the true mean fill weight, and judge whether the machine is meeting its 500 g target.
- Compute the standard error of the mean: σ/√n = 8/√64 = 8/8 = 1 g — averaging 64 bags shrinks bag-to-bag noise of 8 g down to 1 g of uncertainty about the mean.
- Pick the critical value for 95% confidence from the normal distribution: 1.96 standard errors on each side.
- Form the margin of error: 1.96 × 1 = 1.96 g.
- Build the interval: 502 ± 1.96 gives (500.04, 503.96) — the data are consistent with true mean fills between roughly 500.0 and 504.0 grams.
- Draw the inference: the target value 500 sits just outside the interval, so at the 95% level the data suggest the machine is overfilling slightly — a systematic excess of about 2 g per bag, small but real money across thousands of bags.
- Sanity-check the logic: with n = 16 instead of 64, the standard error would be 2 g and the interval 502 ± 3.92 would comfortably contain 500 — the same sample mean would not justify the overfilling conclusion, which shows how sample size, not luck, sharpens inference.
Answer. The 95% confidence interval is (500.04, 503.96) g; since 500 g lies just outside it, the data indicate the machine is overfilling by about 2 g on average.
Check your understanding
- Why does the standard error shrink with √n rather than with n, and what does that imply about the cost of extra precision?
- What exactly does the 95% refer to in a confidence interval, and why is it a statement about the method rather than about μ?
- How would the interval change if σ were unknown and had to be estimated from the sample, and which distribution replaces the normal?
- What trade-off do you accept when moving from 95% to 99% confidence with the same data?
Build the foundations first
Statistical inference & estimation builds on these concepts. If any feel shaky, start there.