Chapter 13.5, Problem 165E

a:

To determine

Testing hypothesis.

Explanation of Solution

Null hypothesis (H₀) is $\left({\text{p}}_1-{\text{p}}_2\right)=0$ and alternate hypothesis is (H₁) $\left({\text{p}}_1-{\text{p}}_2\right)>0$.

The Z table value can be calculated as follows:

$\begin{array}{c}\text{z}>{\text{Z}}_{\alpha }\\ ={\text{Z}}_{0.05}\\ =1.645\end{array}$

The Z table value is 1.645.

The estimated p₁ value can be calculated as follows:

$\begin{array}{c}{\widehat{\text{p}}}_{\text{1}}=\frac{{\text{Event}}_{\text{Occured}}}{{\text{Event}}_{\text{Total}}}\mathbb{k}\\ =\frac{68}{163}\\ =0.4172\end{array}$

The estimated p₂ value can be calculated as follows:

$\begin{array}{c}{\widehat{\text{p}}}_{\text{2}}=\frac{{\text{Event}}_{\text{Occured}}}{{\text{Event}}_{\text{Total}}}\\ =\frac{29}{108}\\ =0.2685\end{array}$

The estimated total pvalue can be calculated as follows:

$\begin{array}{c}\widehat{\text{p}}=\frac{{\text{Event 1}}_{\text{Occured}}+{\text{Event 2}}_{\text{Occured}}}{{\text{Event}}_{\text{Total}}}\\ =\frac{68+29}{163+108}\\ =0.3579\end{array}$

The calculated Z value (z) can be calculated as follows:

$\begin{array}{c}\text{z}=\frac{{\text{(}\widehat{\text{p}}}_{\text{1}}-{\widehat{\text{p}}}_{\text{2}}\text{)}}{\sqrt{\widehat{\text{p}}\text{(1}-\widehat{\text{p}}\text{)}\left(\frac{\text{1}}{{\text{n}}_{\text{1}}}+\frac{\text{1}}{{\text{n}}_{\text{2}}}\right)}}\\ =\frac{(0.4172-0.2685)}{\sqrt{0.3579(1-0.3579)\left(\frac{1}{163}+\frac{1}{108}\right)}}\\ =2.50\end{array}$

The Z value is 2.5.

The p value (p) can be calculated as follows:

$\begin{array}{c}\text{p}=1-\text{P}\left(Z<2.5\right)\\ =1-0.9938\\ =0.0062\end{array}$

The p value is 0.0062. Since the p value is nearlyequal to 0, the alternate hypothesis is accepted.

b:

To determine

Testing hypothesis.

Explanation of Solution

Null hypothesis (H₀) is $\left({\text{p}}_1-{\text{p}}_2\right)=0$ and alternate hypothesis is (H₁) $\left({\text{p}}_1-{\text{p}}_2\right)\ne 0$.

The Z table value can be calculated as follows:

$\begin{array}{c}\text{z}>{\text{Z}}_{\frac{\alpha }{2}}\\ ={\text{Z}}_{\frac{0.05}{2}}\\ =1.96\end{array}$

The Z table value is 1.96.

The estimated p₁ value can be calculated as follows:

$\begin{array}{c}{\widehat{\text{p}}}_{\text{1}}=\frac{{\text{Event}}_{\text{Occured}}}{{\text{Event}}_{\text{Total}}}\mathbb{k}\\ =\frac{20}{54}\\ =0.3704\end{array}$

The estimated p₂ value can be calculated as follows:

$\begin{array}{c}{\widehat{\text{p}}}_{\text{2}}=\frac{{\text{Event}}_{\text{Occured}}}{{\text{Event}}_{\text{Total}}}\sqrt{a^2+b^2}\\ =\frac{10}{23}\\ =0.4348\end{array}$

The estimated total pvalue can be calculated as follows:

$\begin{array}{c}\widehat{\text{p}}=\frac{{\text{Event 1}}_{\text{Occured}}+{\text{Event 2}}_{\text{Occured}}}{{\text{Event}}_{\text{Total}}}\\ =\frac{20+10}{54+23}\\ =0.3896\end{array}$

The calculated Z value (z) can be calculated as follows:

$\begin{array}{c}\text{z}=\frac{{\text{(}\widehat{\text{p}}}_{\text{1}}-{\widehat{\text{p}}}_{\text{2}}\text{)}}{\sqrt{\widehat{\text{p}}\text{(1}-\widehat{\text{p}}\text{)}\left(\frac{\text{1}}{{\text{n}}_{\text{1}}}+\frac{\text{1}}{{\text{n}}_{\text{2}}}\right)}}\\ =\frac{(0.3704-0.4348}{\sqrt{0.3896(1-0.3896)\left(\frac{1}{54}+\frac{1}{23}\right)}}\\ =-0.53\end{array}$

The Z value is -0.53.

The p value (p) can be calculated as follows:

$\begin{array}{c}\text{p}=2\text{P}\left(Z<-0.53\right)\\ =2\left(0.2981\right)\\ =0.5962\end{array}$

The p value is 0.5962.  Since the p value is significantly different from 0, the alternate hypothesis is not accepted.