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Confidence Interval for a Population Proportion

When calculating a confidence interval for a proportion, you should keep the following in mind:
  • In practice, we may trust this confidence interval when the number of successes is $15$ or greater and the number of failures is $15$ or greater. That is, when $np_0 \geq 15$ and $n(1 - p_0) \geq 15$.

  • The confidence levels can be inaccurate unless your sample size sample is large. The actual confidence level can be less than the confidence level you specify.

  • This page also computes the "plus four" interval. Statistical literature suggests that the plus four interval yields better results than the usual large-sample interval. Experts in statistical practice recommend that you use the plus four interval for estimating a proportion when the confidence level at least $90\%$ and the sample size is at least $10.$

  • When the guidelines for this interval are not met, you may use a non-parametric alternative: Bootstrap Interval for a Single Population Proportion.
Number of Successes Sample Size
Sample Data:$k=$ $n=$
Confidence Level:

Sample Size: $n=$
Sample Proportion: $\hat{p}=$
Standard Error: $\mbox{SE}_{\hat{p}}=$
Critical $z$ Value:$z^{*}=$
% Confidence Interval:

Plus Four Confidence Interval:
Plus Four Sample Size: $n+4=$
Plus Four Proportion: $\tilde{p}=$
Plus Four Standard Error: $\mbox{SE}_{\tilde{p}}=$
Critical $z$ Value:$z^{*}=$
% Confidence Interval: