= Consistency identities for a two-sided t-test
{title2=$p=2\{1-F_\nu(|t|)\}$}
= t-test output identities
{synonym}
For one two-sided t procedure with observed estimate $\widehat\Delta$, positive <standard error> $s$, null value $\Delta_0$ and reference <Student t-distribution> with $\nu$ <statistical degrees of freedom>, the <test statistic> and <p-value> must satisfy
$$
t=\frac{\widehat\Delta-\Delta_0}{s},\qquad p=2\{1-F_\nu(|t|)\}.
$$
Its symmetric <confidence interval> at <coverage probability> $1-\alpha$ is $[\widehat\Delta-t_{\nu,1-\alpha/2}s,\widehat\Delta+t_{\nu,1-\alpha/2}s]$ and therefore has midpoint $\widehat\Delta$. When the same <standard error> and reference distribution are used, excluding $\Delta_0$ is equivalent to $|t|>t_{\nu,1-\alpha/2}$ and hence $p<\alpha$ for a continuous reference distribution. This proves the test–<interval> equivalence and supplies several <independent> checks on a printed output. An approximate <Welch's t-test> uses its same approximate <statistical degrees of freedom> throughout. Rounding can blur a boundary decision but cannot explain a confidence-interval midpoint far from the reported estimate.
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