Consider independent random samples from two populations tha

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 ₁ and ₂ are unknown but we have reason to believe that ₁ = ₂, we can pool the standard deviations. Using sample sizes n₁ and n₂, the sample test statistic x₁ x₂ has a Student\'s t distribution where

t =

with degrees of freedom d.f. = n₁ + n₂ 2

and where the pooled standard deviation s is
s =

Note: 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 a report regarding average incidence of fox rabies in two regions. For region I, n₁ = 17, x₁ 4.60, and s₁ = 2.62and for region II, n₂ = 12, x₂ 3.81, and s₂ = 2.25. The two sample standard deviations are sufficiently close that we can assume ₁ = ₂. Use the method of pooled standard deviation to consider the report, testing if there is a difference in population mean average incidence of rabies at the 10% level of significance. (Round your t-value to three decimal places and your P-value to four decimal places.)

x1 x2

s 1 n1 + 1 n2

Solution

S=[((16*6.8644)+(11*5.0625))/27]=6.1303=2.4759, t=[(4.6-3.81)/((2.4759)*((1/17)+(1/12)))=.79/.9334=.8464. And the p value at (17+12-2)=27 degrees of freedom at 10% level of significance is 1.314. Therefore,  there is no difference in population mean average incidence of rabies.