In a study of 

486 comma 260486,260 

cell phone​ users, it was found that 

8989 

developed cancer of the brain or nervous system. Assuming that cell phones have no​ effect, there is a 

0.0002180.000218 

probability of a person developing cancer of the brain or nervous system. We therefore expect about 

107107 

cases of such cancer in a group of 

486 comma 260486,260 

people. Estimate the probability of 

8989 

or fewer cases of such cancer in a group of 

486 comma 260486,260 

people. What do these results suggest about media reports that cell phones cause cancer of the brain or nervous​ system?

To estimate the probability of 89 or fewer cases of brain or nervous system cancer in a group of 486,260 people, given the probability of developing such cancer is 0.000218, we will use the normal approximation to the binomial distribution.

### Step-by-Step Solution

1. **Define the parameters**:
   - Sample size \( n = 486,260 \)
   - Probability of success \( p = 0.000218 \)
   - Number of successes (cancer cases) \( k = 89 \)

2. **Calculate the mean and standard deviation**:
   - Mean \( \mu = np = 486,260 \times 0.000218 \approx 106.99668 \) (approximately 107)
   - Standard deviation \( \sigma = \sqrt{np(1-p)} \approx \sqrt{486,260 \times 0.000218 \times (1 - 0.000218)} \approx 10.344 \)

3. **Set up the normal approximation**:
   - We are looking for the probability of 89 or fewer cases. For the normal approximation, we use a continuity correction by adding 0.5.
   - So, we need \( P(X \leq 89.5) \)

4. **Convert to the standard normal variable**:
   - \( Z = \frac{X - \mu}{\sigma} \)
   - For \( X = 89.5 \):
     \[
     Z = \frac{89.5 - 106.99668}{10.344} \approx -1.685
     \]

5. **Find the probability**:
   - We use the standard normal distribution table to find the area to the left of \( Z = -1.685 \).
   - \( P(Z < -1.685) \approx 0.046 \)

So, the probability of 89 or fewer cases of brain or nervous system cancer in a group of 486,260 people is approximately \( 0.046 \) or \( 4.6\% \).

### Interpretation of the Results

The results suggest that if cell phones had no effect on the probability of developing brain or nervous system cancer, the probability of observing 89 or fewer cases in a sample of 486,260 people is relatively low (4.6%). This suggests that the observed number of 89 cases is lower than what we would expect if cell phones had no effect, which might imply that there is no significant increase in cancer cases due to cell phone usage. Thus, these results suggest that media reports claiming cell phones cause brain or nervous system cancer may not be supported by this data.

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Do you want details or have any questions? Here are some related questions for further exploration:

1. What are the assumptions behind using the normal approximation to the binomial distribution?
2. How does the continuity correction improve the accuracy of the normal approximation?
3. What is the central limit theorem, and how is it applied in this context?
4. How would the results change if the sample size were larger or smaller?
5. What other factors could contribute to the discrepancy between expected and observed cancer cases?
6. How can we determine if there is a causal relationship between cell phone use and cancer?
7. What statistical tests can be used to further investigate the link between cell phone use and cancer?
8. How do we interpret the p-value obtained in this context?

**Tip:** Always check the assumptions of your statistical methods to ensure they are appropriate for your data. This can help avoid misleading conclusions.

Explore how to estimate the probability of brain or nervous system cancer cases in a group of 486,260 people using statistical analysis. Understand the application of binomial distribution and normal approximation in probability theory.

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