11.2: Confidence Interval for the Difference between Two Proportions
- Page ID
- 52007
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)The form of the confidence interval is
\[ (\hat{p}_{1} - \hat{p}_{2}) - E < (p_{1} - p_{2}) < (\hat{p}_{1} - \hat{p}_{2}) + E \]with
\[ E = z_{\cal{C}} \sqrt{\frac{\hat{p}_{1}\hat{q}_{1}}{n_{1}} + \frac{\hat{p}_{2}\hat{q}_{2}}{n_{2}} } \]where, as usual you can get \(z_{\cal{C}}\) from the last line of the t Distribution Table.
Example 11.2 : Using the data from Example 11.1, find the 95\(\%\) confidence interval for \(p_{1} - p_{2}\).
Solution : The relevant numbers from Example 11.1 are: \(n_{1} = 34\), \(\hat{p}_{1} = 0.35\), \(\hat{q}_{1} = 1 - 0.35 = 0.65\) and \(n_{2} = 24\), \(\hat{p}_{2} = 0.71\), \(\hat{q}_{1} = 1 - 0.71 = 0.29\).
Compute (after finding \(z_{95\%} = 1.96\) from the t Distribution Table)
\[\begin{eqnarray*} E & = & z_{95\%} \sqrt{\frac{\hat{p}_{1}\hat{q}_{1}}{n_{1}} + \frac{\hat{p}_{2}\hat{q}_{2}}{n_{2}} }\\ E & = & 1.96 \sqrt{\frac{(0.35)(0.65)}{34} + \frac{(0.71)(0.29)}{24} }\\ E & = & 0.242 \end{eqnarray*}\]and
\[ \hat{p}_{1} - \hat{p}_{2} = 0.35 - 0.71 = -0.36 \]So
\[\begin{eqnarray*} (\hat{p}_{1} - \hat{p}_{2}) - E & < (p_{1} - p_{2}) < & (\hat{p}_{1} - \hat{p}_{2}) + E \\ -0.36 - 0.242 & < (p_{1} - p_{2}) < & -0.36 + 0.242 \\ -0.602 & < (p_{1} - p_{2}) < & -0.118 \end{eqnarray*}\]with 95\(\%\) confidence. (Note that this corresponds with the rejection of \(H_{0}\) in Example 11.1 since 0 is not in the confidence interval.)
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