For patients with a certain disease the amount of certain protein in the blood is normally distributed with a mean of 130 and a standard deviation of 5. For patients without the disease the amount of that protein is normally distributed with a mean of 150 and a standard deviation of 4.
Where would you want the “Tested Positive” to start so that the probability of a person with the disease will not test positive is .0025?
To find the cutoff point for the "Tested Positive" result, we need to calculate the z-score that corresponds to a cumulative probability of 0.0025.
First, let's calculate the z-score for patients without the disease, since we know their mean (150) and standard deviation (4). Using the formula:
z = (x - μ) / σ
For a probability of 0.0025, we need to find the z-score for the lower tail of the distribution. Using a z-table or calculator, we can find that the z-score corresponding to a cumulative probability of 0.0025 is approximately -2.81.
Now, let's find the corresponding protein level for the "Tested Positive" cutoff for patients without the disease:
x = z * σ + μ
x = -2.81 * 4 + 150
x ≈ 139.16
Therefore, if the "Tested Positive" result for patients without the disease starts at 139.16, the probability of a person with the disease not testing positive will be approximately 0.0025.