SECTION 8.2 Summary

• The estimate of the difference in two population proportions is
  

  where

• The standard error of the difference is
  

  and the margin of error for confidence level C is

  where is the value for the standard Normal density curve with area C between and .

• The large-sample level C confidence interval for the difference in two proportions is
  

  We recommend using this method when the number of successes and the number of failures in both samples are at least 10.

• The plus four confidence interval for comparing two proportions is obtained by adding one success and one failure to each sample and then using the procedure. We recommend using this method when both sample sizes are at least 5 and the confidence level is 90%, 95%, or 99%.
• Significance tests of use the statistic
  

  with -values from the distribution. In this statistic,

  where is the pooled estimate of the common value of and ,

  We recommend using this test when the number of successes and the number of failures in each of the samples are at least 5.

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• The sample sizes for each of the two proportions needed for a specified value of the margin of error for the difference in two proportions are
  

  Here is the critical value for confidence C, and and are guessed values for and , the proportions of successes in the future sample.

• The margin of error will be less than or equal to if and are chosen to be 0.5. The common sample size required is then given by
  

• Software can be used to determine the sample size needed to detect a given difference in population proportions by specifying the hypotheses, the two values of the proportions, the type I error, and the power.
• Relative risk is the ratio of two sample proportions:

  Confidence intervals for relative risk are an alternative to confidence intervals for the difference when we want to compare two proportions.