Answer to Question #121051 in Statistics and Probability for reesha segovia

Question #121051
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?
1
Expert's answer
2020-06-09T18:05:48-0400

"\\text{Let the proportion of industrial community is}\\; p_{1}\\\\\n\\text{and the proportion of rural community is}\\; p_{2},\\\\\n\\hat p_{1}=\\frac{80}{150}=0.533\\\\\n\\hat p_{2}=\\frac{30}{100}=0.3\\\\\n\\hat p_{p}=\\frac{80+30}{150+100}=0.44\\\\\nH_0: p_{1}= p_{2}\\\\\nH_1:p_{1}\\neq p_{2}\\\\\nLet \\; \\alpha =0.05\\implies Z_{\\frac{\u03b1}{2}}=Z_{0.025}=1.96,\\\\\nZ=\\frac{\\hat p_{1} -\\hat p_{2}}{\\sqrt{\\hat p_{p}(1-\\hat p_{p})(\\frac{1}{n_{1}}+\\frac{1}{n_{2}})}}\\\\\nZ=\\frac{0.533 -0.3}{\\sqrt{0.44(0.56)(\\frac{1}{150}+\\frac{1}{100})}}=3.64\\\\\n\\text{the rejection region} :\\;z>1.96, z<-1.96\\\\\n\\text{The decision is to reject}:\\; H_0\\\\\n\\text{this means that there is a difference between }\\\\\n\\text{ the proportions of people who suffer from }\\\\\n\\text{the disease in the two communities }"


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