Question

The mayor of a town has proposed a plan for the construction of a new community. A political study took a sample of 900 voters in the town and found that 75% of the residents favored construction. Using the data, a political strategist wants to test the claim that the percentage of residents who favor construction is more than 72%. Testing at the 0.05 level, is there enough evidence to support the strategist's claim?

Answer 1

Answer with explanation:

Let p be the population proportion of residents who favor construction.

As per given , we have

Null hypothesis : $H_0: p\leq0.72$

Alternative hypothesis :  $H_a: p>0.72$

Since $H_a$ is right-tailed , so the hypothesis test is a right-tailed z-test.

Also, it is given that , the sample size : n= 900

Sample proportion: $\hat{p}=0.75$

Test statistic : $z=\dfrac{\hat{p}-p}{\sqrt{\dfrac{p(1-p)}{n}}}$ , where n is sample size ,  $\hat{p}$ is sample proportion and p is the population proportion.

$\Rightarrow\ z=\dfrac{0.75-0.72}{\sqrt{\dfrac{0.72(1-0.72)}{900}}}\approx2$

P-value (right tailed test)=P(z>2)=1-P(z≤2)     [∵P(Z>z)=1-P(Z≤z)]

$=1-0.9772=0.0228$   [using p-value table of z.]

Decision : Since P-value (0.0228) < Significance level (0.05), so we reject the null hypothesis .

Thus , we concluded that we have enough evidence at 0.05 significance level to support the strategist's claim that the percentage of residents who favor construction is more than 72%.