SECTION 8.2 Summary

• The estimate of the difference in two population proportions is
  

  $D={\hat{p}}_1-{\hat{p}}_2$

  where

  $\begin{array}{lll}{\hat{p}}_1={\displaystyle \frac{X_1}{n_1}}& \text{and}& {\hat{p}}_2={\displaystyle \frac{X_2}{n_2}}\end{array}$

• The standard error of the difference is
  

  ${\text{SE}}_D=\sqrt{{\displaystyle \frac{{\hat{p}}_1\left(1-{\hat{p}}_1\right)}{n_1}}+{\displaystyle \frac{{\hat{p}}_2\left(1-{\hat{p}}_2\right)}{n_2}}}$

  and the margin of error for confidence level C is

  $m=z^{*}{\text{SE}}_D$

  where $z^{*}$ is the value for the standard Normal density curve with area C between $-z^{*}$ and $z^{*}$.

• The $z$ large-sample level C confidence interval for the difference in two proportions $p_1-p_2$ is
  

  $\left({\hat{p}}_1-{\hat{p}}_2\right)\pm m$

  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 $z$ 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 $H_0:p_1=p_2$ use the $z$ statistic
  

  $z={\displaystyle \frac{{\hat{p}}_1-{\hat{p}}_2}{{\text{SE}}_{Dp}}}$

  with $P$-values from the $N(0,1)$ distribution. In this statistic,

  ${\text{SE}}_{Dp}=\sqrt{\hat{p}\left(1-\hat{p}\right)\left({\displaystyle \frac{1}{n_1}}+{\displaystyle \frac{1}{n_2}}\right)}$

  where $\hat{p}$ is the pooled estimate of the common value of $p_1$ and $p_2$,

  $\hat{p}={\displaystyle \frac{X_1+X_2}{n_1+n_2}}$

  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 $m$ of the margin of error for the difference in two proportions are
  

  $n={\left({\displaystyle \frac{z^{*}}{m}}\right)}^2\left(p_1^{*}\left(1-p_1^{*}\right)+p_2^{*}\left(1-p_2^{*}\right)\right)$

  Here $z^{*}$ is the critical value for confidence C, and $p_1^{*}$ and $p_2^{*}$ are guessed values for $p_1$ and $p_2$, the proportions of successes in the future sample.

• The margin of error will be less than or equal to $m$ if $p_1^{*}$ and $p_2^{*}$ are chosen to be 0.5. The common sample size required is then given by
  

  $n=\left({\displaystyle \frac{1}{2}}\right){\left({\displaystyle \frac{z^{*}}{m}}\right)}^2$

• 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:

  $\text{RR}={\displaystyle \frac{{\hat{p}}_1}{{\hat{p}}_2}}$

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