Racial Crossover. In the paper "The Racial Crossover in Comorbidity, Disability, and Mortality" (Demography, Vol. 37(3), pp. 267-283), N. Johnson investigated the health of independent random samples of white and African-American elderly (aged 70 years or older). Of the 4989 white elderly surveyed, 529 had at least one stroke, whereas 103 of the 906 African-American elderly surveyed - Lported at least one stroke. At the $5\%$significance level, do the data suggest that there is a difference in stroke incidence between white and African-American elderly?

The given values are,

$$\begin{gathered} x_1=529, \\ {n}_1=4989, \\ {x}_2=103, \\ {n}_2=906, \\ \alpha /2=0.025. \end{gathered}$$

The formula for ${\tilde{p}}_1$is given by,

${\tilde{p}}_1=\frac{x_1}{n_1}$

The formula for ${\tilde{p}}_2$is given by,

${\tilde{p}}_2=\frac{x_2}{n_2}$

The formula for $z$is given by,

localid="1651312298549" $z=\frac{{\tilde{p}}_1-{\tilde{p}}_2}{\sqrt{{\tilde{p}}_p\left(1-{\tilde{p}}_p\right)}\sqrt{\frac{1}{n_1}+\frac{1}{n_2}}}$

The formula for ${\tilde{p}}_p$is given by,

${\tilde{p}}_p=\frac{x_1+x_2}{n_1+n_2}$

The value of ${\tilde{p}}_1$is calculated as,

$$\begin{gathered} {\tilde{p}}_1=\frac{x_1}{n_1} \\ =\frac{529}{4989} \\ =0.1060 \end{gathered}$$

The value of ${\tilde{p}}_2$is calculated as,

$$\begin{gathered} {\tilde{p}}_2=\frac{x_2}{n_2} \\ =\frac{103}{906} \end{gathered}$$

$=0.1137$

The value of ${\tilde{p}}_p$is calculated as,

${\tilde{p}}_p=\frac{x_1+x_2}{n_1+n_2}$

$$\begin{gathered} =\frac{529+103}{4989+966} \\ =0.1072 \end{gathered}$$

The value of $z$is calculated as,

$z=\frac{{\tilde{p}}_1-{\tilde{p}}_2}{\sqrt{\tilde{p}\left(1-{\tilde{p}}_p\right)}\sqrt{\frac{1}{n_1}+\frac{1}{n_2}}}$

$$\begin{gathered} =\frac{0.1060-0.1137}{\sqrt{0.1172(1-0.1172)}\left(\sqrt{\frac{1}{4989}+\frac{1}{906}}\right)} \\ =-0.69 \end{gathered}$$

Calculate two-tailed critical values are.

$$\begin{gathered} \pm z_{a/2}=\pm z_{0.05/2} \\ =\pm z_{0.025} \\ =\pm 1.96 \end{gathered}$$

$\left|z\right|=|-0.69|=0.69$the test statistic values less than the critical value of$1.96.$

$$\begin{gathered} {\tilde{p}}_2=\frac{x_2}{n_2} \\ =\frac{103}{906} \end{gathered}$$

We concluded that there is no difference in stroke incidence between African - the American elderly at the $5\%$level of significance.

As a result, at a 5% level of significance, the data do not suggest a difference in stroke incidence between and African - the American elderly.