The question is:
In a study of 420,000 cell phone users in Denmark. It found that 135 developed cancer of the brain or nervous system. If we assume that such a cancer is not affected by cell phone, the probability of a person having such a cancer is 0.000340.
a) Assuming that cell phones have no affect on cancer, find the mean and tandard deviation for the numbers of people in groups of 420,000 that can be expected to have cancer of the brain or nervous system.
b) Based on the results from part (a), it is unusual to find that among 420,000 people, there were 135 cases of cancer of the brain or nervous system? Why ot why not?
c) What do the results suggest about the publicized concern that cell phones are a health danger because they increase the risk of cancer of the brain or nervous system?
a) To find the mean (μ) and standard deviation (σ), we need to use the formula for the binomial distribution:
μ = n * p
σ = √(n * p * (1-p))
Where:
n = number of trials (420,000 in this case)
p = probability of success (0.000340 in this case)
Substituting the values into the formulas:
μ = 420,000 * 0.000340 = 142.8
σ = √(420,000 * 0.000340 * (1-0.000340)) = 11.75
Therefore, the mean number of people expected to have cancer of the brain or nervous system in groups of 420,000 is approximately 142.8, and the standard deviation is approximately 11.75.
b) To determine if it is unusual to find 135 cases of cancer among 420,000 people, we can use the concept of z-score. The z-score is calculated using the formula:
z = (x - μ) / σ
Where:
x = observed number of cases (135 in this case)
μ = mean (142.8)
σ = standard deviation (11.75)
Substituting the values:
z = (135 - 142.8) / 11.75 ≈ -0.66
We can now find the corresponding p-value using a standard normal distribution table or a statistical software. A p-value represents the probability of observing a value as extreme as, or more extreme than, the observed value. If the p-value is below a certain threshold (e.g., 0.05), we can conclude that the observation is statistically unusual.
c) The p-value represents the evidence against the null hypothesis that cell phones have no effect on cancer. If the p-value is less than the threshold (e.g., 0.05), it suggests that the observed number of cases (135) is statistically significant, and there is evidence to suggest that cell phones might increase the risk of cancer of the brain or nervous system.
However, if the p-value is greater than the threshold, it suggests that the observed number of cases is not statistically significant, and there is insufficient evidence to conclude that cell phones are a health danger in increasing the risk of cancer.