Hypothesis Tests

Two-Sample t-Test (pooled)

Compares the means of two independent samples assuming equal variances.

Statusvalidated

Version1.0.0

Minitab equivalentStat > Basic Statistics > 2-Sample t (Assume equal variances)

Description

The two-sample t-test (pooled) tests whether two independent, normally distributed populations have the same mean. It assumes equal variances (homogeneity of variance) — for unequal variances, the Welch t-test should be used. The pooled variance sp² combines the variability of both samples.

Formulas

t = \frac{x ˉ _{1} - x ˉ _{2}}{s _{p} \frac{1}{n _{1}} + \frac{1}{n _{2}}}

Test statistic with pooled standard deviation

sp²

s_{p}^{2} = \frac{( n _{1} - 1 ) s _{1}^{2} + ( n _{2} - 1 ) s _{2}^{2}}{n _{1} + n _{2} - 2}

Pooled variance: weighted average of sample variances

H₀

H_{0} : μ_{1} = μ_{2}

Null hypothesis: The population means are equal

Assumptions

Both samples are normally distributed
Samples are independent
Variances are equal (homogeneity of variance)

Limitations

Use Welch t-test for unequal variances
Sensitive to outliers

References

Student (W. S. Gosset) (1908), The probable error of a mean, Biometrika, 6(1), 1–25
NIST/SEMATECH e-Handbook of Statistical Methods, Section 7.3.1