Calculated result
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Test formulas
One-sample t = (x̄ − μ₀)/(s/√n), with df = n − 1. Welch’s test uses SE = √(s₁²/n₁ + s₂²/n₂) and the Welch–Satterthwaite fractional degrees of freedom.
Worked example
For x̄ = 12, μ₀ = 10, s = 2, and n = 25, t = 5, df = 24, and the two-sided p-value is about 0.00004157.
Local p-values and boundaries
Student t probabilities use a bounded regularized incomplete-beta calculation. One-sided direction follows the displayed difference. No significance verdict, causation claim, or study-quality judgment is made.