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.

Question

(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?

Answer 1

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.