For each successive presidential term from Franklin Pierce (the 14th
president, elected in 1853 ) to George W. Bush (43rd president), the party
affiliation controlling the White House is shown below, where R designates
Republican and D designates Democrat (Reference: The New York Times Almanac).
Historical Note: We start this sequence with the 14 th president because
earlier presidents belonged to political parties such as the Federalist or
Wigg (not Democratic or Republican) party. In cases in which a president died
in office or resigned, the period during which the vice president finished the
term is not counted as a new term. The one exception is the case in which
Lincoln (a Republican) was assassinated and the vice president Johnson (a
Democrat) finished the term.
$$
\begin{array}{llllllllllllllllllll}
\text { D } & \text { D } & \text { R } & \text { R } & \text { D } & \text {
R } & \text { R } & \text { R } & \text { R } & \text { D } & \text { R } &
\text { D } & \text { R } & \text { R } & \text { R } & \text { R } & \text {
D } & \text { D } & \text { R } & \text { R } \\
\text { D } & \text { D } & \text { D } & \text { D } & \text { D } & \text {
R } & \text { R } & \text { D } & \text { D } & \text { R } & \text { R } &
\text { D } & \text { R } & \text { R } & \text { R } & \text { D } & \text {
D } & \text { R } & &
\end{array}
$$
Test the sequence for randomness at the \(5 \%\) level of significance. Use the
following outline.
(a) State the null and alternate hypotheses.
(b) Find the number of runs \(R, n_{1}\), and \(n_{2} .\) Let \(n_{1}=\) number of
Republicans and \(n_{2}=\) number of Democrats.
(c) In this case, \(n_{1}=21\), so we cannot use Table 10 of Appendix II to find
the critical values. Whenever either \(n_{1}\) or \(n_{2}\) exceeds 20, the number
of runs \(R\) has a distribution that is approximately normal, with
$$
\mu_{R}=\frac{2 n_{1} n_{2}}{n_{1}+n_{2}}+1 \text { and } \quad
\sigma_{R}=\sqrt{\frac{\left(2 n_{1} n_{2}\right)\left(2 n_{1}
n_{2}-n_{1}-n_{2}\right)}{\left(n_{1}+n_{2}\right)^{2}\left(n_{1}+n_{2}-1\right)}}
$$
We convert the number of runs \(R\) to a \(z\) value, and then use the normal
distribution to find the critical values. Convert the sample test statistic
\(R\) to \(z\) using the formula
$$
z=\frac{R-\mu_{R}}{\sigma_{R}}
$$
(d) The critical values of a normal distribution for a two-tailed test with
level of significance \(\alpha=0.05\) are \(-1.96\) and \(1.96\) (see Table
\(5(\mathrm{c})\) of Appendix II. Reject \(H_{0}\) if the sample test statistic \(z
\leq-1.96\) or if the sample test statistic \(z \geq 1.96\). Otherwise, do not
reject \(H_{0}\).
$$
\begin{array}{c|c|c}
\text { Sample } z \leq-1.96 & -1.96<\text { sample } z<1.96 & \text { Sample
} z \geq 1.96 \\
\hline \text { Reject } H_{0} & \text { Fail to reject } H_{0} . & \text {
Reject } H_{0}
\end{array}
$$
Using this decision process, do you reject or fail to reject \(H_{0}\) at the \(5
\%\) level of significance? What is the \(P\) -value for this two-tailed test? At
the \(5 \%\) level of significance, do you reach the same conclusion using the
\(P\) -value that you reach using critical values? Explain.
(e) Interpret your results in the context of the application.
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