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Difference of Successes


If x_1/n_1 and x_2/n_2 are the observed proportions from standard normally distributed samples with proportion of success theta, then the probability that

 w=(x_1)/(n_1)-(x_2)/(n_2)
(1)

will be as great as observed is

 P_delta=1-2int_0^(|delta|)phi(t)dt,
(2)

where

delta=w/(sigma_w)
(3)
sigma_w=sqrt(theta^^(1-theta^^)(1/(n_1)+1/(n_2)))
(4)
theta^^=(x_1+x_2)/(n_1+n_2).
(5)

Here, theta^^ is the unbiased estimator. The skewness and kurtosis excess of this distribution are

gamma_1^2=((n_1-n_2)^2)/(n_1n_2(n_1+n_2))(1-4theta^^(1-theta^^))/(theta^^(1-theta^^))
(6)
gamma_2=(n_1^2-n_1n_2+n_2^2)/(n_1n_2(n_1+n_2))(1-6theta^^(1-theta^^))/(theta^^(1-theta^^)).
(7)

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Cite this as:

Weisstein, Eric W. "Difference of Successes." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/DifferenceofSuccesses.html

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