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: $(0.04,\ 0.21)$

Step-by-step explanation:

The confidence interval for population proportion is given by :-

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

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

Critical value : $z_{\alpha/2}=2.576$

$n_1=400 ;\ p_1=\dfrac{160}{400}=0.4$

$n_2=400 ;\ p_2=\dfrac{110}{400}=0.275$

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

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

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