Question

A sample of 150 people from a certain industrial community showed that 80 people suffered from a lung disease. A sample of 100 people from a rural community showed that 30 suffered from the same lung disease. At α =0.05, is there a difference between the proportions of people who suffer from the disease in the two communities?

Answer 1

$\text{Let the proportion of industrial community is}\; p_{1}\\ \text{and the proportion of rural community is}\; p_{2},\\ \hat p_{1}=\frac{80}{150}=0.533\\ \hat p_{2}=\frac{30}{100}=0.3\\ \hat p_{p}=\frac{80+30}{150+100}=0.44\\ H_0: p_{1}= p_{2}\\ H_1:p_{1}\neq p_{2}\\ Let \; \alpha =0.05\implies Z_{\frac{α}{2}}=Z_{0.025}=1.96,\\ Z=\frac{\hat p_{1} -\hat p_{2}}{\sqrt{\hat p_{p}(1-\hat p_{p})(\frac{1}{n_{1}}+\frac{1}{n_{2}})}}\\ Z=\frac{0.533 -0.3}{\sqrt{0.44(0.56)(\frac{1}{150}+\frac{1}{100})}}=3.64\\ \text{the rejection region} :\;z>1.96, z<-1.96\\ \text{The decision is to reject}:\; H_0\\ \text{this means that there is a difference between }\\ \text{ the proportions of people who suffer from }\\ \text{the disease in the two communities }$