REFERENCE SHEET

Confidence intervals & tests

Express sampling uncertainty without confusing the data with the hypothesis.

Before this LLN, CLT & concentration, Estimation & likelihood

\bar{X} \pm z_{1 - α / 2} \frac{σ}{\sqrt{n}} normal data, known σ

Definitions

Confidence procedure

A random interval C(X) with coverage Pθ(θ∈C(X))≥1−α for the specified model and parameter range.

Null hypothesis and p-value

A p-value is, under the null model, the probability of a test statistic at least as extreme as the observed one in the specified tail or tails.

Type I error and power

Type I error rejects a true null. Power is the rejection probability at a specified alternative.

Key theorems

Normal mean interval

For iid N(μ,σ²) data with known σ, the displayed interval has exact coverage 1−α. With unknown σ and n≥2, replace σ by s and the normal quantile by a tₙ₋₁ quantile.

Union bound for testing

Testing m true null hypotheses each at level α/m gives probability of at least one false rejection at most α, without an independence assumption.

Notable applications

Experiments

Intervals show plausible effect sizes and precision; tests quantify incompatibility with a chosen null.

Multiple comparisons

Multiplicity corrections help control error when examining many hypotheses.

Work it out

Try before revealing

01 / Compute

For normal data with known σ=10, n=100, and X̄=50, give a 95% confidence interval for μ.

Hint

Use z₀.₉₇₅≈1.96.

Solution

The standard error is 1, so the interval is [48.04,51.96].

02 / Compute

What per-test level gives a Bonferroni family-wise level of 0.05 for 20 tests?