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

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 =
`(25)/(100) * 160`

=
`40`

So,
`n_1=160 , y_1=40`

You sample 120 women, and 20% own cats.

No. of women have cats =
`(20)/(100) * 120`

=
`24`

So,
`n_2=120 , y_2=24`

We will use Comparing Two Proportions

`\widehat{p_1}=(y_1)/(n_1)`

`\widehat{p_1}=(40)/(160)`

`\widehat{p_1}=0.25`

`\widehat{p_2}=(y_2)/(n_2)`

`\widehat{p_2}=(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<p_2`

`\widehat{p}=(y_1+y_2)/(n_1+n_2) =(24+40)/(160+120)=0.228`

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

Substitute the values

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

test statistic :
`0.986`

So, test statistic is 0.986