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Population Comparison

Let X_1 and X_2 be the number of successes in variates taken from two populations. Define

p^^_1=(x_1)/(n_1)
(1)
p^^_2=(x_2)/(n_2).
(2)

The estimator of the difference is then p^^_1-p^^_2. Doing a so-called z-transform,

z=((p^^_1-p^^_2)-(p_1-p_2))/(sigma_(p^^_1-p^^_2)),
(3)

where

sigma_(p^^_1-p^^_2)=sqrt(sigma_(p^^_1)^2-sigma_(p^^_2)^2).
(4)

The standard error is

SE_(p^^_1-p^^_2)=sqrt((p^^_1(1-p^^_1))/(n_1)+(p^^_2(1-p^^_2))/(n_2))
(5)
SE_(x^__1-x^__2)=sqrt((s_1^2)/(n_1)+(s_2^2)/(n_2))
(6)
s_(pool)^2=((n_1-1)s_1^2+(n_2-1)s_2^2)/(n_1+n_2-2).
(7)

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References

Gonick, L. and Smith, W. The Cartoon Guide to Statistics. New York: Harper Perennial, pp. 162-171, 1993.

Referenced on Wolfram|Alpha

Population Comparison

Cite this as:

Weisstein, Eric W. "Population Comparison." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/PopulationComparison.html

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