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Under the condition that both populations have equal standard deviations (σ1 - σ2), we can pool the standard deviations and use a Student s t distribution with degrees of freedom d.f. - n1 + n2 - 2 to fi nd the margin of error of a c confi dence interval for µ1 - µ2. This technique demonstrates another commonly used method of computing confi dence intervals for µ1 - µ2.
PROCEDURE
How to find a confidence interval for µ1 - µ2 when σ1 - σ2
Requirements
Consider two independent random samples, where
x̅1 and x̅2 are sample means from populations 1 and 2
s1 and s2 are sample standard deviations from populations 1 and 2
n1 and n2 are sample sizes from populations 1 and 2
If you can assume that both population distributions 1 and 2 are normal or at least mound-shaped and symmetric, then any sample sizes n1 and n2 will work.
If you cannot assume this, then use sample sizes n1 ≥ 30 and n2 ≥ 30.
Confidence interval for µ1 - µ2 when σ1 - σ2
Where
(pooled standard deviation)
c = confidence level (0 < c < 1)
tc - critical value for confi dence level c and degrees of freedom
d.f. = n1 + n2 - 2
With statistical software, select 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. The pooled standard deviation method applies even if the standard deviations are known to be only approximately equal. Consider Problem 23 regarding weights of gray |
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