Question

It is estimated that approximately 8.3% Americans are afflicted with diabetes. Suppose that a certain diagnostic evaluation for diabetes will correctly diagnose 94.5% of all adults over 40 with diabetes as having the disease and incorrectly diagnoses 2.5% of all adults over 40 without diabetes as having the disease. a) Find the probability that a randomly selected adult over 40 does not have diabetes, and is diagnosed as having diabetes

Answer 1

The probability that a randomly selected adult over 40 does not have diabetes and is diagnosed as having diabetes (false positive) is 2.2925%.

Step-by-step explanation:

To find the probability that a randomly selected adult over 40 does not have diabetes but is diagnosed as having diabetes, we can use the given statistics:

• 8.3% of Americans are afflicted with diabetes (diabetic rate).
• 2.5% of all adults over 40 without diabetes are incorrectly diagnosed as having the disease (false positive rate).

The probability that an adult over 40 does not have diabetes is simply 100% - 8.3% = 91.7%.

To find the probability of a false positive:

1. First, calculate the probability that an adult over 40 does not have diabetes, P(No Diabetes) = 91.7%.
2. Next, use the false positive rate of 2.5% to determine the probability of being incorrectly diagnosed.
3. The desired probability is the product of the two: P(False Positive) = P(No Diabetes) * Probability of Incorrect Diagnosis.

Calculating this, we get:

P(False Positive) = 0.917 * 0.025 = 0.022925, or 2.2925%.