Cell Phones and Brain Cancer In a study of 420,095 cell phone ...

Question

Cell Phones and Brain Cancer In a study of 420,095 cell phone users in Denmark, it was found that 135 developed cancer of the brain or nervous system. For those not using cell phones, there is a 0.000340 probability of a person developing cancer of the brain or nervous system. We therefore expect about 143 cases of such cancers in a group of 420,095 randomly selected people.
a. Find the probability of 135 or fewer cases of such cancers in a group of 420,095 people.
b. What do these results suggest about media reports that suggest cell phones cause cancer of the brain or nervous system?

a. Find the probability of 135 or fewer cases of such cancers in a group of 420,095 people.

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

Accepted Answer

Hi everyone. Let's take a look at this practice problem. This problem says in a study of 420,0095 bottled water samples tested over a period of time, 135 were found to have lead levels exceeding safety limits. For water samples drawn from untreated sources, the probability of detecting such high lead levels is 0.000340. So in a group of 420,000, 95 randomly selected water samples, we expect around 143 to show excessive lead levels. There are two parts to this problem. For part one, what is the probability of the of observing 135 or fewer contaminated samples out of the 420,000, 95. And for part two, based on this, what can we infer about the claims that bottled water frequently contains harmful lead concentrations? We're also given four possible choices as our answers. For choice A, for Part one, we have 0.907, and for part two, the data supports the claim. For choice B, for part 1, we have 0.032. And for part two, the data do not support the claim. For choice C for part one, we have 0.480, and for part two, the data supports the claim. And for choice D for part 1, we have 0.268. And for part 2, the data do not support the claim. So, for part one, we need to calculate the probability of observing 135 or fewer contaminated samples out of the 420,0095 that we have. So in this problem we're dealing with a binomial distribution with a large number of trials. So we're going to use a Poisson approximation, and recall that the Poisson approximation is used when the number of trials is large and the probability of success is very small. So the first thing we need to do is to calculate the mean number of expected cases, and we'll label that as lambda. And we're actually given this information in the problem since we're told that we expect around 143 to show excessive lead levels. So that means that lambda, It's going to be equal to 143. So next we need to calculate the Poisson cumulative probability using our formula. So recall for that formula? We have P X. Less than or equal to K. is equal to the sum. Of X equal to 02 K. Of the minus lambda, multiplied by lambda to the X. And that is all divided by x factorum. So here, K is going to be the number of bottles that we found in our samples. That's going to be the 135. So those are the number of bottles that contained lead. So we're going to set K equal to 135 and calculate this probability. So P of capital X is less than or equal to 135. is going to be equal to, and here, using our calculator, we get a value of 0.268. So this year is going to be our answer for part one. Now for part two, We need to answer the question, what can we infer about this claim that bottled water frequently contains harmful lead concentrations? So since the probability of getting 135 or fewer contaminated samples is around 26.8%, that is not unusually low. So, the observer results are consistent with natural random variation. So that means that our data do not Support the claim. That's gonna be our answer for part two. And if we take our answers here and compare them to our answer choices, we see that the correct answer choice corresponds to answer choice D. So, I hope this explanation has been useful, and I'll see you in the next video.

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Key Concepts

Probability Distribution

Hypothesis Testing

Statistical Significance

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