Question

You are testing the claim that the proportion of men who own cats is smaller than the proportion of women who own cats. You sample 160 men, and 25% own cats. You sample 120 women, and 20% own cats. Find the test statistic, rounded to two decimal places.

Answer 1

Answer:

Given:You sample 160 men, and 25% own cats

          You sample 120 women, and 20% own cats.

To Find : Find the test statistic, rounded to two decimal places.

Solution:

You sample 160 men, and 25% own cats.

No. of men have cats = $\frac{25}{100} \times 160$

                                    = $40$

So, $n_1=160 , y_1=40$

You sample 120 women, and 20% own cats.

No. of women have cats = $\frac{20}{100} \times 120$

                                    = $24$

So, $n_2=120 , y_2=24$

We will use Comparing Two Proportions

$\widehat{p_1}=\frac{y_1}{n_1}$

$\widehat{p_1}=\frac{40}{160}$

$\widehat{p_1}=0.25$

$\widehat{p_2}=\frac{y_2}{n_2}$

$\widehat{p_2}=\frac{24}{120}$

$\widehat{p_2}=0.2$

Let $p_1$ and $p_2$ be the probabilities of men having cat and women having cat receptively

$H_0:p_1=p_2\\H_a:p_1$

$\widehat{p}=\frac{y_1+y_2}{n_1+n_2} =\frac{24+40}{160+120}=0.228$

Formula of test statistic : $\frac{\widehat{p_1}-\widehat{p_2}}{\sqrt{\widehat{p}(1-\widehat{p})(\frac{1}{n_1}+\frac{1}{n_2})}}$

Substitute the values  

test statistic : $\frac{0.25-0.2}{\sqrt{0.228(1-0.228)(\frac{1}{160}+\frac{1}{120})}}$

test statistic : $0.986$

So, test statistic is 0.986