Estatística t (1 amostra)

One-sample t-test

The one-sample t-statistic checks the sample mean against a hypothesized value in the situation where you don't know the population standard deviation: t = (x̄ − μ₀) / (s / √n), with s standing in for the sample standard deviation. Student's t-distribution carries heavier tails than the normal and is governed by the degrees of freedom df = n − 1; as n → ∞ it settles into the standard normal. You then compare |t| to the critical value t_crit pulled from a t-table, and that critical value drifts toward 1.96 as df grows. Take x̄ = 102, μ₀ = 100, s = 8, n = 30: that works out to t ≈ 1.37 with df = 29, so you don't reject H₀ at 5%.

Applications

Think of lab experiments running on small samples (n between 5 and 30), method validation in analytical chemistry, or checking a sample mean against a theoretical or specified target. It also shows up in clinical trials with modest sample sizes and in acceptance testing whenever the historical σ can't be trusted.

FAQ

Why use t instead of z? Because s only estimates σ, and that estimate is shaky when n is small. The heavier tails of t are what absorb that extra uncertainty.

What is df = n − 1? Those are the degrees of freedom. You lose one because s gets computed from the sample mean itself. The more df you have, the tighter and more normal-like the distribution becomes.

Does t-test require normality? With small n you really want it, ideally. Once n > 30, the central limit theorem keeps the test robust against mild departures from normality.