The National Association of Realtors (NAR) reported the results of an April 2015 survey of home buyers. In a random sample of 1,971 residential properties purchased during the year, 414 were purchased as a vacation home. Five years ago, 10% of residential properties were vacation homes.

a. Do the survey results allow the NAR to conclude (at $\mathit{\alpha }\mathbf{=}\mathbf{.}\mathbf{01}$) that the percentage of all residential properties purchased for vacation homes is greater than 10%?

b. In a previous year, the NAR sent the survey questionnaire to a nationwide sample of 45,000 new home owners, of which 1,982 responded to the survey. How might this bias the results? [Note: In the most recent survey, the NAR used a more valid sampling method.

According to the National Association of Realtors, out of 1,971 residential properties purchased during 2015, 414 were purchased as vacation homes.

That is

The size of the sample$n=1971$

The sample proportion is

$$\begin{gathered} \hat{p}=\frac{414}{1971} \\ =0.210 \end{gathered}$$

Where $\hat{p}$is the sample proportion of residential properties purchased for vacation homes.

Here we have to test whether the true population proportion of residential properties purchased for vacation homes is greater than 10% or not.

The null and alternative hypotheses are given as

$H_0:p=0.10$

That is, there is no statistical evidence that the true population proportion of residential properties purchased for vacation homes is greater than 10%.

And

$H_a:p>0.10$

That is, there is statistical evidence that the true population proportion of residential properties purchased for vacation homes is greater than 10%.

The test statistic for testing these hypotheses is

$$\begin{gathered} Z=\frac{\hat{p}-p}{\sqrt{\frac{p\left(1-p\right)}{n}}} \\ =\frac{0.210-0.10}{\sqrt{\frac{0.10\left(1-0.10\right)}{1971}}} \\ =\frac{0.11}{\sqrt{0.00004566}} \\ =16.28 \end{gathered}$$

Here

$\alpha :$The level of significance

$\alpha =.01$

Using the standard normal table, the critical value at the 1% significance level is 2.326

We can see that

$Z=16.28>2.326$

Hence, we reject the null hypothesis.

At a 1% significance level, we have sufficient evidence to conclude that the true percentage of all residential properties purchased for vacation homes is greater than 10%.

Since NAR randomly sent the survey questionnaire to the 45,000 new homeowners, only 1,982 responded. Approximately 95% of owners just ignored the survey. This is called the non-response bias in a statistical term.

Hence, the survey result cannot be reliable and does not represent the true proportion of residential properties purchased for vacation homes.