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?
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.
To determine if the number of people with toxic levels of serum alpha tocopherol is unusual, we can calculate the probability using the normal distribution.
Step 1: Calculate the z-score
The z-score is calculated using the formula:
z = (x - μ) / σ
Given that x = 2000 μg/dL, μ = 860 μg/dL, and σ = 340 μg/dL, we can calculate the z-score:
z = (2000 - 860) / 340
z = 1.176
Step 2: Find the probability
We need to find the probability of observing a z-score of 1.176 or higher. We can use a standard normal distribution table or a calculator to find this probability.
Using a standard normal distribution table, the probability of observing a z-score of 1.176 or higher is 0.8790.
Step 3: Interpret the probability
The probability of observing a z-score of 1.176 or higher is 0.8790, which means that the chance of having serum alpha tocopherol levels above 2000 μg/dL in a sample of 2000 people is 0.8790 or 87.90%.
Since the probability is relatively high (87.90%), it indicates that having 4 people with serum alpha tocopherol levels above 2000 μg/dL in a sample of 2000 people is not unusual.