Question

A random sample of 400 items from a population shows that 160 of the sample items possess a given characteristic. A random sample of 400 items from a second population resulted in 110 of the sample items possessing the characteristic. Using this data, a 99% confidence interval is constructed to estimate the difference in population proportions which possess the given characteristic. The resulting confidence interval is _______.

Answer 1

Answer:
`(0.04,\ 0.21)`

Explanation:

The confidence interval for population proportion is given by :-

`p_1-p_2\pm z_(\alpha/2)\sqrt{(p_1(1-p_1))/(n_1)+(p_2(1-p_2))/(n_2)}`

Given : Significance level :
`\alpha=1-0.99=0.01`

Critical value :
`z_(\alpha/2)=2.576`

`n_1=400 ;\ p_1=(160)/(400)=0.4`

`n_2=400 ;\ p_2=(110)/(400)=0.275`

Then a 99% confidence interval is constructed to estimate the difference in population proportions which possess the given characteristic will be :-

`0.4-0.275\pm (2.576)\sqrt{(0.4(1-0.4))/(400)+(0.275(1-0.275))/(400)}\\\\\approx0.125\pm0.085=(0.125-0.085,0.125+0.085)=(0.04,\ 0.21)`

Hence, the resulting confidence interval is
`(0.04,\ 0.21)` .