. 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 determine the threshold at which a "Tested Positive" result should start in order to achieve a probability of .0025 for a person with the disease not testing positive, we need to find the corresponding z-score in the normal distribution.

1. First, we need to find the z-score for a probability of .0025 in a standard normal distribution (mean = 0, standard deviation = 1). This can be done using a z-table or a statistical calculator. In this case, the z-score is approximately -2.81.

2. Next, we need to convert this z-score to the corresponding value in our specific normal distribution with a mean of 130 and a standard deviation of 5. Let's call this value x.

The formula to convert a z-score to a raw score is:
x = z * standard deviation + mean

In this case:
x = -2.81 * 5 + 130
x = -14.05 + 130
x ≈ 115.95

Therefore, the "Tested Positive" threshold should start at approximately 115.95 for a person with the disease, so that the probability of them not testing positive is approximately .0025.

To find the value at which "Tested Positive" should start so that the probability of a person with the disease not testing positive is 0.0025, we need to calculate the z-score using the formula:

z = (x - μ) / σ

where:
- z is the z-score representing the number of standard deviations away from the mean
- x is the value we are interested in (the starting point for "Tested Positive")
- μ is the mean of the population (130 for patients with the disease)
- σ is the standard deviation of the population (5 for patients with the disease)

Now we can rearrange the formula to solve for x:

x = z * σ + μ

Since the probability of a person with the disease not testing positive is 0.0025, we are interested in the z-score that corresponds to a cumulative probability of 0.0025 in the standard normal distribution table.

Looking up this value in the table, we find that the z-score is approximately -2.81.

Plugging in the values into the formula, we can calculate the starting point for "Tested Positive":

x = (-2.81) * 5 + 130
x = -14.05 + 130
x ≈ 115.95

Therefore, to ensure that the probability of a person with the disease not testing positive is 0.0025, "Tested Positive" should start at approximately 115.95.