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? Use any method.

Answer 1

Given: x1= 80 , n1=150,

x2= 30, n2=100

Step1: state the hypothesis

H0: $p1=p2$

H1: $p1\neq p2$

Step2: Name of the test:

2 sample proportion test

Step 3: test statistic:

$\widehat{p1}=\frac{x_{1}}{n_{1}}=\frac{80}{150}=0.5333, \widehat{p2}=\frac{x_{2}}{n_{2}}=\frac{30}{100}=0.3, \widehat{p}=\frac{x_{1}+x_{2}}{n_{1+}n_{2}}=\frac{80+30}{150+100}=0.44$

Test statistic is calculated as follows:

$z=\frac{\widehat{p1}-\widehat{p2}}{\sqrt{\widehat{p}(1-\widehat{p})(\frac{1}{n_{1}}+\frac{1}{n_{2}})}}$

$z=\frac{0.5333-0.3}{\sqrt{0.44(1-0.44)(\frac{1}{150}+\frac{1}{100})}}$

=3.641

Step 4: P value Using z table : We get a z-score of 3.641. Because this is a two-sided test, it is not enough to just look at the right tail. We also have to look at the equivalent of the left tail, or a -3.641. that is P(Z<-3.641)+P(Z>=3.641)=0.000136+0.000136=0.000272=0.0003

Step 5: conclusion: since P value is less than alpha , null hypothesis is rejected.

Hence,there is enough evidence to claim that there is difference between proportions of people suffer from the disease in two communities.