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Confidence Interval Comparing Two Population Proportions

When calculating a confidence interval involving proportions, you should keep the following in mind:
  • The confidence levels can be very inaccurate unless your sample size sample is very large. The actual confidence level is often less than the confidence level you specify. Use this interval when in both samples there are 10 or more successes (i.e., $n_1\hat{p}_1\geq 10$ and $n_2 \hat{p}_2\geq 10$) and 10 or more failures (i.e., $n_1(1-\hat{p}_1)\geq 10$ and $n_2(1-\hat{p}_2)\geq 10$).

  • This website also computes the "plus four" interval. Statistical literature suggests that the plus four interval yields better results than the usual large-sample interval. Experts recommend that you use the plus four interval for estimating the difference of two proportions. The plus four interval can be used when both samples have at least 5 data points.

  • When the above guidelines for this interval are not met, you may use a non-parametric alternative: Bootstrap Interval for the Difference of Two Population Proportions.
Number of Successes Sample Size $n$
Sample 1: $k_1=$ $n_1=$
Sample 2:$k_2=$ $n_2=$
Confidence Level:

Sample Sizes: $n_1=$ $n_2=$
Sample Proportions: $\hat{p}_1=$ $\hat{p}_2=$
Difference Estimate: $\hat{p}_1-\hat{p}_2=$
Standard Error: $\mbox{SE}_{\hat{p}_1-\hat{p}_2}=$
Critical z Value:$z^{*}=$
% Confidence Interval:

Plus Four Confidence Interval:
Plus Four Sample Sizes: $n_1+2=$ $n_2+2=$
Plus Four Proportions: $\tilde{p}_1=$ $\tilde{p}_2=$
Difference Estimate: $\tilde{p}_1-\tilde{p}_2=$
Plus Four Standard Error: $\mbox{SE}_{\tilde{p}_1-\tilde{p}_2}=$
Critical z Value:$z^{*}=$
% Confidence Interval: