Section 10.2 Summary of Comparing Two Population Proportions
Parameter: \(\pi_1 - \pi_2\) = the difference in the population proportions of success
To test \(H_0: \pi_1 - \pi_2 = 0\)
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Simulation: Random samples from binomial processes with common \(\pi\)
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Two-sample z-test:The standardized statistic \(z_0 = \frac{\hat{p}_1 - \hat{p}_2}{\sqrt{\hat{p}(1-\hat{p})(\frac{1}{n_1} + \frac{1}{n_2})}}\) where \(\hat{p} = \frac{X_1+X_2}{n_1+n_2} = \frac{\text{total number of successes}}{\text{total number in study}}\)is well approximated by the standard normal distribution when \(n_1\hat{p} > 5\text{,}\) \(n_1(1 - \hat{p}) > 5\text{,}\) \(n_2\hat{p} > 5\) and \(n_2(1 - \hat{p}) > 5\text{,}\) where \(\hat{p}\) is the proportion of successes combining the 2 groups.
Approximate (100 × C)% Confidence interval for \(\pi_1 - \pi_2\text{:}\)
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Two-sample z-interval:An approximate (100 × C)% interval: \((\hat{p}_1 - \hat{p}_2) \pm z^* \sqrt{\frac{\hat{p}_1(1-\hat{p}_1)}{n_1} + \frac{\hat{p}_2(1-\hat{p}_2)}{n_2}}\)This method is considered valid when \(n_1\hat{p}_1 > 5\text{,}\) \(n_1(1 - \hat{p}_1) > 5\text{,}\) \(n_2\hat{p}_2 > 5\text{,}\) and \(n_2(1 - \hat{p}_2) > 5\text{.}\)
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Or the Wilson or Wald adjustment can be made on the sample proportions and sample sizes first.
Note: We can simplify the technical conditions by verifying that you have independent random samples from two processes or large populations (or one random sample classified by a binary variable) and at least 5 successes and 5 failures in each group.
Technology: To perform the calculations for these two-sample z-procedures you can use
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R:
iscamtwopropztest -
JMP: e.g., Fit Y by X or ISCAM Journal file > Hypothesis Test for Two Proportions and Confidence Interval for Two Proportions
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