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 x̅₁ ­– x̅₂ has a Student’s t distribution, where

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

With statistical software, select the pooled variance or equal variance options.

(a) 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 23 regarding average incidence of fox rabies in two regions. For region I, n₁ = 16, x̅₁ = 4.75, and s₁ ≈ 2.82 and for region II, n₂ = 15, x̅₂ ≈ 3.93, and s₂ ≈ 2.43. The two sample standard deviations are suffi­ciently close that we can assume σ₁ = σ₂. Use the method of pooled standard deviation to redo Problem 23, where we tested if there was a difference in population mean average incidence of rabies at the 5% level of signifi­cance.

(b) Compare the t value calculated in part (a) using the pooled standard deviation with the t value calculated in Problem 23 using the unspooled standard deviation. Compare the degrees of freedom for the sample test statistic. Compare the conclusions.

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1 n2 V n1