Question

Critical Thinking. In Exercises 17–28, use the data and confidence level to construct a confidence interval estimate of p, then address the given question.Cell Phones and Cancer A study of 420,095 Danish cell phone users found that 0.0321% of them developed cancer of the brain or nervous system. Prior to this study of cell phone use, the rate of such cancer was found to be 0.0340% for those not using cell phones. The data are from the Journal of the National Cancer Institute.

a. Use the sample data to construct a 90% confidence interval estimate of the percentage of cell phone users who develop cancer of the brain or nervous system.

b. Do cell phone users appear to have a rate of cancer of the brain or nervous system that is different from the rate of such cancer among those not using cell phones? Why or why not?

Short answer

a.The 90% confidence interval is equal to (0.0276%,0.0366%).

b. No, cell phone users do not appear to have a rate of cancer of the brain or nervous system that is different from the rate of such cancer among those not using cell phones.

Step-by-step solution

Given Information

In a sample of 420095 Danish cell phone users, 0.0321% of them developed cancer of the brain or nervous system. The rate of cancer was found to be 0.0340% for those not using cell phones.

Calculation of the sample proportion

The sample proportion of cell phone users who developed cancer is computed below:

\(\begin{array}{c}\hat p = 0.0321\% \\ = \frac{{0.0321}}{{100}}\\ = 0.000321\end{array}\)

The sample proportion of cell phone users who did not develop cancer is computed below:

\(\begin{array}{c}\hat q = 1 - \hat p\\ = 1 - 0.000321\\ = 0.999679\end{array}\)

Calculation of the margin of error

a.

The given level of significance is 0.10

Therefore, the value of\({z_{\frac{\alpha }{2}}}\)from the standard normal table is equal to 1.645.

The margin of error is computed as shown:

\(\begin{array}{c}E = {z_{\frac{\alpha }{2}}} \times \sqrt {\frac{{\hat p\hat q}}{n}} \\ = 1.645 \times \sqrt {\frac{{0.000321 \times 0.999679}}{{420095}}} \\ = 0.0000455\end{array}\)

Therefore, the margin of error is 0.0000455.

Calculation of the confidence interval

a.

The 90% confidence interval is computed as follows:

\(\begin{array}{c}\hat p - E < p < \hat p + E\\0.000321 - 0.0000455 < p < 0.000321 + 0.0000455\\0.000276 < p < 0.000367\\0.0276\% < p < 0.0367\% \end{array}\)

Thus, the 90% confidence interval is equal to (0.0276%,0.0367%).

Step 5:Conclusion 

b.

Since confidence interval includes the value of 0.034%, there does not appear to be any difference betweenthe rate of cancer of the brain or nervous system that develops in cell phone users as compared to the rate of such cancer among those not using cell phones.