Compare means when the population standard deviation is unknown. Supports one-sample, two-sample (Welch's), and paired designs. Exact p-values from the t-distribution.
One-sample t statistic:
t = (xΜ β ΞΌβ) / (s / βn)
Degrees of freedom:
df = n β 1
95% CI for ΞΌ:
xΜ Β± t*(Ξ±/2, n-1) Γ (s / βn)
Welch's t-test (unequal variances):
t = (xΜβ β xΜβ) / β(sβΒ²/nβ + sβΒ²/nβ)
Welch-Satterthwaite degrees of freedom:
df = (sβΒ²/nβ + sβΒ²/nβ)Β² / [(sβΒ²/nβ)Β²/(nβ-1) + (sβΒ²/nβ)Β²/(nβ-1)]
Pooled t-test (equal variances):
spΒ² = [(nβ-1)sβΒ² + (nβ-1)sβΒ²] / (nβ+nβ-2)
t = (xΜβ β xΜβ) / (sp Γ β(1/nβ + 1/nβ))
Compute differences dα΅’ = xβα΅’ β xβα΅’, then:
t = dΜ / (sd / βn)
Degrees of freedom:
df = n β 1
CI for mean difference:
dΜ Β± t*(Ξ±/2, n-1) Γ (sd / βn)