Question

A poll was conducted between town voters and country voters for favoring the certain proposal. If 60 of 100 town voters favor to proposal and 150 of 300 country residents favor it, would you agree that the proportion of town voters favoring the proposal is higher than the proportion of country voters at 5% level of significance?.

Answer 1

$\text{Let the proportion of town voter is}\; p_{1}\\ \text{and the proportion of country voter is}\; p_{2},\\ \hat p_{1}=\frac{60}{100}=0.6\\ \hat p_{2}=\frac{150}{300}=0.5\\ \hat p_{p}=\frac{60+150}{100+300}=0.525\\ H_0: p_{1}\leq p_{2}\\ H_1:p_{1}> p_{2}\\ Let \; \alpha =0.05\implies Z_{α}=Z_{0.05}=1.65,\\ Z=\frac{\hat p_{1} -\hat p_{2}}{\sqrt{\hat p_{p}(1-\hat p_{p})(\frac{1}{n_{1}}+\frac{1}{n_{2}})}}\\ Z=\frac{0.6 -0.5}{\sqrt{0.525(0.475)(\frac{1}{100}+\frac{1}{300})}}=1.73\\ \text{the rejection region} :\;z>1.65\\ \text{The decision is to reject}:\; H_0\\ \text{this means that the proportion of town voters is }\\ \text{higher than the proportion of country voters }$