Question

1.The distribution of serum levels of alpha tocopherol (serum vitamin E) is here assumed to be distributed normal with mean μ=860 μg/dL and standard deviation σ=340 μg/dL.
(a).A study is undertaken for evidence of toxicity among 2000 people who regularly take vitamin-E supplements. The investigators found that 4 people have serum alpha tocopherol levels > 2000 μg/dL. Is this an unusual number of people with toxic levels of serum alpha tocopherol?

Answers

GPT 3.5
To determine if the number of people with toxic levels of serum alpha tocopherol is unusual, we need to calculate the probability of observing 4 or more people with levels > 2000 μg/dL in a sample of 2000.

First, we need to convert the values to z-scores using the formula:
z = (x - μ) / σ

For a serum level of 2000 μg/dL:
z = (2000 - 860) / 340
z = 1140 / 340
z ≈ 3.35

Next, we can calculate the probability using a standard normal distribution table or a calculator:

P(Z > 3.35) ≈ 0.0004

So, the probability of finding an individual with a serum level > 2000 μg/dL is approximately 0.0004.

To find the probability of observing 4 or more people with levels > 2000 μg/dL in a sample of 2000, we can use the binomial distribution formula:

P(X ≥ 4) = 1 - P(X ≤ 3)

Where X follows a binomial distribution with n = 2000 and p ≈ 0.0004.

Using a calculator or a computer program, we find:

P(X ≥ 4) ≈ 1 - binomcdf(2000, 0.0004, 3)
≈ 1 - 0.107
≈ 0.893

Therefore, the probability of observing 4 or more people with toxic levels of serum alpha tocopherol in a sample of 2000 is approximately 0.893.

Since this probability is relatively high (close to 1), it suggests that having 4 people with levels > 2000 μg/dL is not an unusual occurrence.

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