Question

In 2003, 6 percent of vacation-home owners were minorities. The National Association of Realtors (NAR) reported the results of a May 2006 survey of vacation-home owners. In a sample of 416 households that owned one or more vacation homes, 46 were minorities. Do the survey results allow the NAR to conclude at a 0.01 level of significance that the percentage of vacation-home owners who are minorities is larger than 6 percent?

Answer 1

Answer with explanation:

Let p represents the population proportion.

From the given information, we have

`H_0: p=0.06\\\\H_a: p>0.06`, since alternative hypothesis is right tailed then the test is right tailed test.

Given : In a sample of 416 households that owned one or more vacation homes, 46 were minorities.

i.e. n= 416

Sample proportion :
`\hat{p}=(46)/(416)=0.110576923077\approx0.11`

Test statistic :
`z=\frac{\hat{p}-p}{\sqrt{(p(1-p))/(n)}}`

i.e.
`z=\frac{0.11-0.06}{\sqrt{(0.06(1-0.06))/(416)}}`

`\Rightarrow\ z=4.29414907893\approx4.29`

P-value for right tailed test :
`P(z>4.29)=1-P(z\leq4.29)`

`=1-0.999991=0.0000089`

Since the p-value is less than the significance level (0.01), so we reject the null hypothesis.

Thus , we conclude that we have enough evidence to support the claim that that the percentage of vacation-home owners who are minorities is larger than 6 percent.