Question

Ballistic Fingerprinting. Refer to Exercise 11.110 and find and interpret a $98\%$confidence interval for the difference between the percentages of women and men who favor ballistic fingerprinting.

Short answer

There is $98\%$interval for the difference between the proportion is (0.147,0.273)

Step-by-step solution

Given Information

The given values are,

$$\begin{gathered} x_1=446, \\ {n}_1=537, \\ {x}_2=307, \\ {n}_2=495, \\ \alpha =0.02 \end{gathered}$$

Explanation

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,

$z=\frac{{\tilde{p}}_1-{\tilde{p}}_2}{\sqrt{{\tilde{p}}_p\left(1-{\tilde{p}}_p\right)}\sqrt{\frac{1}{m_1}+\frac{1}{m_2}}}$

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

$$\begin{gathered} {\tilde{p}}_1=\frac{x_1}{n_1} \\ =\frac{446}{537} \\ =0.83 \end{gathered}$$

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

$$\begin{gathered} {\tilde{p}}_2=\frac{x_2}{n_2} \\ =\frac{307}{495} \\ =0.62 \end{gathered}$$

Calculate the value of $z_{\frac{a}{2}}$with $98\%$confidence level is.

$z\frac{a}{2}=Z0.2/2$

$$\begin{gathered} =Z0.01 \\ =2.326 \end{gathered}$$

The $98\%$confidence interval is.

$\left({\tilde{p}}_1-{\tilde{p}}_2\right)\pm z_{\frac{a}{2}}\sqrt{\frac{{\tilde{p}}_1\left(1-{\tilde{p}}_1\right)}{n_1}+\frac{{\tilde{p}}_1\left(1-{\tilde{p}}_1\right)}{n_2}}=(0.83-0.62)\pm 2.326\sqrt{\frac{0.83(1-0.83)}{537}+\frac{0.62(1-0.62)}{495}}$

$=0.21\pm 0.063$

The interval for the difference between the proportion is $(0.147,0.273)$