Question

Fear of Heights Among readers of a USA Today website, 285 chose to respond to this posted question: “Are you afraid of heights in tall buildings?” Among those who chose to respond, 46% answered “yes” and 54% answered “no.” Use a 0.05 significance level to test the claim that the majority of the population is not afraid of heights in tall buildings. What is wrong with this hypothesis test?

Short answer

There is not enough evidence to support the claim that most of the population is not afraid of heights in tall buildings.

The sample is a voluntary response sample,so the test results may not be reliable.

Step-by-step solution

Given information

In a sample size of 285 respondents, 46% said they were afraid of heights in tall buildings, while 54% said that they were not.

The significance level is 0.05.

The sample size (n) is 285.

Identify the hypothesis

The z-test is used to test whether the given proportion is equal to the hypothesized value.

The null hypothesis is as follows:

An equalproportion of respondents answered “yes” and “no” to the question.

\({H_0}:p = 0.5\)

The alternative hypothesis is as follows:

The proportion of respondents whoanswered “no” is greater than the proportion of respondents who responded “yes”.

\({H_1}:p > 0.5\)

It is a right-tailed test.

Test statistic

Here, pis the proportion of people who responded “no”. Thus, p=0.50.

And

\(\begin{array}{c}q = 1 - p\\ = 1 - 0.50\\ = 0.50.\end{array}\)

The estimated value of p is shown below:

\(\begin{array}{c}\hat p = 54\% \\ = \frac{{54}}{{100}}\\ = 0.54\end{array}\)

The sample size (n) is 285.

The value of the test statistic is computed below:

\(\begin{array}{c}z = \frac{{\hat p - p}}{{\sqrt {\frac{{pq}}{n}} }}\\ = \frac{{0.54 - 0.50}}{{\sqrt {\frac{{0.50\left( {1 - 0.50} \right)}}{{285}}} }}\\ = 1.35\end{array}\)

The value of the z score is 1.35.

The critical value of z at \(\alpha = 0.05\) for a right-tailed test is 1.645.

Since the test statistic value is less than the critical value, the null hypothesis fails to reject.

Conclusion

There is not enough evidence to conclude that most of the population is afraid of heights in tall buildings.

Pitfall of the test

Because respondents decided to answer, it is a voluntary response sample and not a random sample.Thus,the results may not be reliable.