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A 100(1−α)% CI for μ: X̄±z_{α/2}(σ/√n) when σ known, or X̄±t_{n-1,α/2}(s/√n) when unknown. Interpretation: 100(1−α)% of repeated CIs contain μ, NOT P(μ in CI)=1−α (μ is fixed, CI is random). t-distribution used because s estimates σ.
From the distribution of the sample mean, you know that if X₁, ..., Xₙ are i.i.d. with mean μ and standard deviation σ, then X̄ is approximately normal with mean μ and standard error σ/√n. From z-scores, you know how to standardize: Z = (X̄ − μ)/(σ/√n) ~ N(0,1). A confidence interval for μ reverses this: instead of computing a probability given μ, you construct a random interval that captures μ with specified probability.
Start with the case where σ is known. Since Z ~ N(0,1), you know P(−z_{α/2} ≤ Z ≤ z_{α/2}) = 1 − α, where z_{α/2} is the value cutting off area α/2 in each tail. Substitute Z = (X̄ − μ)/(σ/√n) and rearrange to isolate μ: P(X̄ − z_{α/2}·σ/√n ≤ μ ≤ X̄ + z_{α/2}·σ/√n) = 1 − α. The interval [X̄ ± z_{α/2}·σ/√n] is the z-interval. For 95% confidence, z_{α/2} ≈ 1.96, giving roughly X̄ ± 2 standard errors. The margin of error σ/√n shrinks as n grows — more data means a tighter interval, as expected.
When σ is unknown (the realistic case), you replace it with the sample standard deviation s. This changes the distribution: the quantity (X̄ − μ)/(s/√n) follows a t-distribution with n−1 degrees of freedom, not a standard normal. The t-distribution is symmetric and bell-shaped like the normal but has heavier tails, especially when n is small, reflecting the additional uncertainty from estimating σ. As n increases, the t-distribution approaches N(0,1), and the t-interval approaches the z-interval. The t-interval [X̄ ± t_{n-1, α/2}·s/√n] is the correct formula for practice whenever σ is unknown.
The most critical conceptual point: the confidence level describes the *procedure*, not any specific computed interval. Once you observe data and compute, say, [3.1, 4.7], the parameter μ either is or is not in that interval — there is no probability about it. "95% confidence" means that if you repeated the entire process (new sample, new CI) many times, 95% of the resulting intervals would contain μ. The interval is the random object; μ is fixed. Holding this picture clearly — the interval moves across repetitions, μ stays put — prevents the most common misinterpretation and builds the right foundation for hypothesis testing.