Equality of the population means

Equality of the population means of two normally distributed populations based on independent random samples with equal variance and unequal variance

With equal variance
Hypothesis
H0: u1 = u2
H1: u1 ≠ u2

Where:
u1 is the population mean of population 1
u2 is the population mean of population 2

Use t-test
Test-statistic = (X1-X2)/Standard error

Standard error = (s²/n1 + s²/n2)^1/2
Estimated population variance: s2 = [(n1-1)s1² – (n2-2)s2²]/(n1+n2-2)

Critical value: look up df= n1+n2-2, p=significance

Where:
n1, n2 are samples sizes,
X1, X2 are sample means
s1², s2² are sample variances.

With unequal variance
Hypothesis
H0: u1 = u2
H1: u1 ≠ u2

Where:
u1 is the population mean of population 1
u2 is the population mean of population 2

Use t-test
Test-statistic = (X1-X2)/Standard error

Standard error = (s1²/n1 + s2²/n2)^1/2

Critical value: look up df=[ (s1²/n1+s2²/n2)²]/{( s1²/n1)²/n1+( s2²/n2)²/n2}, p=significance

Where:
n1, n2 are samples sizes
X1, X2 are sample means
s1², s2² are sample variances