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.

Cell Phones and Cancer In a study of 420,095 Danish cell phone users, 135 subjects developed cancer of the brain or nervous system (based on data from the Journal of the National Cancer Institute as reported in USA Today). Test the claim of a somewhat common belief that such cancers are affected by cell phone use. That is, test the claim that cell phone users develop cancer of the brain or nervous system at a rate that is different from the rate of 0.0340% for people who do not use cell phones. Because this issue has such great importance, use a 0.005 significance level. Based on these results, should cell phone users be concerned about cancer of the brain or nervous system?

Short answer

Null hypothesis: The proportion of cell phone users who develop cancer is equal to 0.000340.

Alternative hypothesis: The proportion of cell phone users who develop cancer is not equal to 0.000340.

Test statistic: -0.667

Critical value: 2.807

P-value: 0.5047

The null hypothesis is failed to reject.

There is not enough evidence to support the claim that the proportion of cell phone users who develop cancer of the brain or nervous system is not equal to 0.0340%.

As the results indicate that the proportion of cell phone users and non-cell phone users who develop cancer is roughly the same, cell phone users should not be concerned.

Step-by-step solution

Given information

Out of 420095 cell phone users, 135 develop cancer of the brain or nervous system. It is claimed that the rate at which cell phone users develop cancer is not equal to 0.0340%.

Hypotheses

The null hypothesis is written as follows.

The proportion of cell phone users who develop cancer is equal to 0.0340%.

$H_0:p=0.000340$

The alternative hypothesis is written as follows.

The proportion of cell phone users who develop cancer is not equal to 0.0340%.

$H_1:p\ne 0.000340$

The test is two-tailed.

Sample size, sample proportion, and population proportion

The sample size is equal to n=420095.

The sample proportion of cell phone users who develop cancer is computed below.

$$\begin{gathered} \hat{p}=\frac{\text{Number}\text{of}\text{users}\text{who}\text{developed}\text{cancer}}{\text{Sample}\text{Size}} \\ =\frac{135}{420095} \\ =0.000321 \end{gathered}$$

The population proportion of cell phone users who develop cancer is equal to 0.000340.

Test statistic

The value of the test statistic is computed below.

$$\begin{gathered} z=\frac{\hat{p}-p}{\sqrt{\frac{pq}{n}}} \\ =\frac{0.000321-0.000340}{\sqrt{\frac{0.000340\left(1-0.000340\right)}{420095}}} \\ =-0.667 \end{gathered}$$

Thus, z=-0.667.

Critical value and p-value

Referring to the standard normal table, the critical value of z at for a two-tailed test is equal to 2.807.

Referring to the standard normal table, the p-value for the test statistic value of -0.667 is equal to 0.5047.

Asthe p-value is greater than 0.005, the null hypothesis is failed to reject.

Conclusion of the test

There is not enough evidence to support the claim that the proportion of cell phone users who develop cancer of the brain or nervous system is not equal to 0.0340%.

As the results suggest that the proportion of cell phone users, as well as non-cell phone users who develop cancer, is approximately the same, cell phone users should not be concerned.