Question

Testing Claims About Proportions. In Exercises 9–32, test the given claim. Identify the null hypothesis, alternative hypothesis, test statistic, P-value, or critical value(s), then state the conclusion about the null hypothesis, as well as the final conclusion that addresses the original claim. Use the P-value method unless your instructor specifies otherwise. Use the normal distribution as an approximation to the binomial distribution, as described in Part 1 of this section.

Is Nessie Real? This question was posted on the America Online website: Do you believe the Loch Ness monster exists? Among 21,346 responses, 64% were “yes.” Use a 0.01 significance level to test the claim that most people believe that the Loch Ness monster exists. How is the conclusion affected by the fact that Internet users who saw the question could decide whether to respond?

Short answer

Null hypothesis: The proportion of people who believe that the Loch Ness monster exists is equal to 50%.

Alternative hypothesis: The proportion of people who believe that the Loch Ness monster exists is greater than 50%.

Test statistic: 40.909

Critical value: 2.3263

P-value: 0.000

The null hypothesis is rejected.

There is enough evidence to support the claim that the proportion of people who believe that the Loch Ness monster exists is greater than 50%.

The sample is a voluntary-response sample and not a simple random sample. Thus, the results of the test maybe inaccurate.

Step-by-step solution

Given information

In a survey involving 21346 people, 64% believe that the Loch Ness monster exists. It is claimed that most people believe that the Loch Ness monster exists.

Hypotheses

The null hypothesis is written as follows.

The proportion of people who believe that the Loch Ness monster exists is equal to 50%.

$H_0:p=0.5$

The alternative hypothesis is written as follows.

The proportion of people who believe that the Loch Ness monster exists is greater than 50%.

$H_1:p>0.5$

The test is right-tailed.

Sample size, sample proportion,and population proportion

The sample size is equal to n=21346.

The sample proportion of people who believe that the Loch Ness monster exists is equal to

$$\begin{gathered} \hat{p}=64\% \\ =\frac{64}{100} \\ =0.64 \end{gathered}$$

The population proportion of people who believe that the Loch Ness monster exists is equal to 0.5.

Test statistic

The value of the test statistic is computed below.

$$\begin{gathered} z=\frac{\hat{p}-p}{\sqrt{\frac{pq}{n}}} \\ =\frac{0.64-0.5}{\sqrt{\frac{0.5\left(1-0.5\right)}{21346}}} \\ =40.909 \end{gathered}$$

Thus, z=40.909.

Critical value and p-value

Referring to the standard normal table, the critical value of z at $\alpha =0.01$ for a right-tailed test is equal to 2.3263.

Referring to the standard normal table, the p-value for the test statistic value of 40.909is equal to 0.000.

As the p-value is less than 0.01, the null hypothesis is rejected.

Conclusion of the test

There is enough evidence to support the claim that the proportion of people who believe that the Loch Ness monster exists is greater than 50%.

If the internet users have chosen to respond to the question, the sample is a voluntary-response sample and cannot be considered a simple random sample.

Thus, the results of the test cannot be relied upon and maybe false.