Question

Two different simple random samples are drawn from two different populations. The first sample consists of 20 people with 9 having a common attribute. The second sample consists of 2100 people with 1492 of them having the same common attribute. Compare the results from a hypothesis test of p 1equalsp 2 ​(with a 0.05 significance​ level) and a 95​% confidence interval estimate of p 1minusp 2.

Answer 1

Answer: $(-0.48,\ -0.04)$

Step-by-step explanation:

The confidence interval for the difference of two 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 : The first sample consists of 20 people with 9 having a common attribute.

Here, $n_1=20$ , $p_1=\dfrac{9}{20}=0.45$

$n_2=2100$ , $p_1=\dfrac{1492 }{2100}\approx0.71$

Significance level : $\alpha=1-0.95=0.05$

Critical value : $z_{\alpha/2}=1.96$

A 95% confidence interval for the difference of two population proportion will be :-

$(0.45-0.71)\pm z_{\alpha/2}\sqrt{\dfrac{0.45(1-0.45)}{20}+\dfrac{0.71(1-0.71)}{2100}}\\\\\approx -0.26\pm0.22\\\\=(-0.26-0.22,-0.26+0.22)\\\\=(-0.48,\ -0.04)$