\setcounter{ExampleCounter}{1}
First of all, to conduct the test for proportions, we use what's called a \textbf{pooled proportion}:
\[p_{pooled} = \dfrac{x_1 + x_2}{n_1 + n_2}\]
\paragraph{Step 1:} State the hypotheses.
\begin{center}
\begin{tabular}{c | c}
$H_0$ & $H_1$\\
\hline
& \\
$p_1 = p_2$ & $p_1 \neq p_2$\\
$p_1 \leq p_2$ & $p_1 > p_2$\\
$p_1 \geq p_2$ & $p_1 < p_2$
\end{tabular}
\end{center}
\paragraph{Step 2:} Calculate the test statistic.
\[z = \dfrac{(\hat{p_1}-\hat{p_2})-(p_1-p_2)}{\sqrt{p_{pooled}(1-p_{pooled})\left(\frac{1}{n_1}+\frac{1}{n_2}\right)}}\]
\paragraph{Step 3:} Calculate the p value.
\begin{center}
\texttt{normalcdf(-1000000,z,0,1)} or similar
\end{center}
\paragraph{Step 4:} Draw a conclusion.
\begin{itemize}
\item If $p < \alpha$, reject $H_0$.
\item If $p > \alpha$, fail to reject $H_0$.
\end{itemize}
\subsection{Using Your Calculator}
Use the \texttt{2-PropZTest} in the \texttt{TESTS} menu. All you have to enter is \texttt{x} and \texttt{n} for each group; make sure to keep the two groups straight. Then enter the appropriate alternate hypothesis and select either \texttt{Calculate} or \texttt{Draw}.
\begin{center}
\includegraphics[width=3in]{Calc2PropZTest}
\end{center}
\begin{example}{Childhood Obesity}
The National Health and Nutrition Examination Survey (NHANES) weighed a sample of 546 boys aged 6--11 and found that 87 of them were overweight. They weighed a sample of 508 girls aged 6--11 and found that 74 of them were overweight. Can you conclude that the proportion of boys who are overweight differs from the proportion of girls who are overweight?
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\end{example}
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\begin{example}{Pollution and Altitude}
In a random sample of 340 cars driven at low altitudes, 46 of them exceeded a standard of 10 grams of particulate pollution per gallon of fuel consumed. In an independent random sample of 85 cars driven at high altitudes, 21 of them exceeded the standard. Can you conclude that the proportion of high-altitude vehicles exceeding the standard is greater than the proportion of low-altitude vehicles exceeding the standard? Use the $\alpha=0.01$ level of significance.
\vspace*{7in}
\end{example}
\vfill
\pagebreak
\begin{example}{Preventing Heart Attacks}
Medical researchers performed a comparison of two drugs, clopidogrel and ticagrelor, which are designed to reduce the risk of heart attack or stroke in coronary patients. A total of 6676 patients were given clopidogrel, and 6732 were given ticagrelor. Of the clopidogrel patients, 668 suffered a heart attack or stroke within one year, and of the ticagrelor patients, 569 suffered a heart attack or stroke. Can you conclude that the proportion of patients suffering a heart attack or stroke is less for ticagrelor? Use the $\alpha=0.01$ level.
\vspace*{7in}
\end{example}
\vfill
\pagebreak