Consider independent random samples from two populations that are normal or approximately normal, or the case in which both sample sizes are at least 30. Then, if σ1 and σ2 are unknown but we have reason to believe that σ1 = σ2, we can pool the standard deviations. Using sample sizes n1 and n2, the sample test statistic 1 - 2 has a Student€™s t distribution, where

with degrees of freedom d.f. = n1 + n2 - 2
and the pooled standard deviation s is

Hypothesis tests: Use H0: μ1 = μ2 and an appropriate alternate hypothesis. The test statistic is

with d.f. = n1 + n2 - 2
A c confidence interval:

where

tc = critical value for confidence level c and degrees of freedom d.f. = n1 + n2 - 2
With statistical software, select the pooled variance or equal variance options.
There are many situations in which we want to compare means from populations having standard deviations that are equal. This method applies even if the standard deviations are known to be only approximately equal. Consider Problem 19 regarding average incidence of fox rabies in two regions. For region I, n1 = 16, 1 = 4.75, and s1 ‰ˆ 2.82, and for region II, n2 = 15, 2 ‰ˆ 3.93, and s2 ‰ˆ 2.43. The two sample standard deviations are sufficiently close that we can assume σ1 = σ2.
(a) Use the method of pooled standard deviation to redo Problem 19(a).
(b) Use the method of pooled standard deviation to redo Problem 19(b).

x1x2 n1 2 1snz