Under the condition that both populations have equal standard deviations ( 1 2 ), we can pool the standard deviations and use a

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 fi­nd the margin of error of a c confi­dence interval for µ₁ ­– µ₂. This technique demonstrates another commonly used method of computing confi­dence 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

t_c – critical value for confi­dence 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 wolves in two regions.

Notice that s₁ = 8.32 pounds and s₂ = 8.87 pounds are fairly close. Use the method of pooled standard deviation to fi­nd an 85% confi­dence interval for the difference in population mean weights of gray wolves in the Chihuahua region compared with those in the Durango region.

(b) Compare the con­fidence interval computed in part (a) with that computed in Problem 23. Which method has the larger degrees of freedom? Which method has the longer confi­dence interval?

n2 +1n2-2