Hypothesis Testing with the t Statistic — interactive tool

Significance ⇄ confidence interval

Significance and the confidence interval say the same thing two ways — and “significant” is not the same as “large”.

Take one regression slope β̂ with standard error se. The t-statistic t = β̂ / se lands on the t-density: if it falls in a red rejection tail we reject H₀: β = 0. The same evidence is the interval β̂ ± t*·se below. The two always agree: the interval excludes 0 ⇔ the test rejects.

95% CI [0.143, 0.757]contains 0? no

Reject H₀: β = 0 at α = 0.05

The CI excludes 0, so the test rejects. Same conclusion.

t = β̂ / se3.000

p-value (two-tail)0.006

critical t*2.048

df = n − k28

CI low0.143

CI high0.757

Magnitude ≠ significance: t depends on β̂ relative tose. A tiny β̂ with an even smaller se is “significant” yet economically trivial. Always read the CI width, not just the verdict.

Coefficient β̂0.45

Std. error se0.150

Sample size n30

Regressors k (incl. intercept)2

Significance α0.05

Try this

Discussion.Load “Significant but tiny”: the estimate clears every significance test yet the whole interval sits between 0.01 and 0.03. Would you call that effect important? Now argue the opposite case from “Big but imprecise”. What does each one tell you that a lone p-value hides?

Two-sided test of H₀: β = 0 with t ~ Student-t(df = n − k). The p-value and t* come from the Student-t distribution (regularized incomplete beta), exact to textbook precision. The CI is β̂ ± t*·se; it excludes 0 exactly when p < α.

Hypothesis Testing with the t StatisticOpen in Dr Phil's Quant Lab ↗