Question

4. A test for diabetes is a measurement X of a person's blood sugar level following an overnight fast. For a healthy person, a blood sugar level X in the range of 70 110 mg/dl is considered normal. When a measurement X is used as a test for diabetes, the result is called positive (event 7+) if X ≥ 140, the test is negative (event T^{-})if~X\le110, and the test is ambiguous (event 7") if 110< X < 140. Given that a person is healthy (event H), a blood sugar measurement X is a Gaussian N(90,20) random variable. Given that a person has diabetes, (event D). X is (160, 40) P(H)=0 random variable. A randomly chosen person is healthy with probability or has diabetes with probability P(D)=0.1.\underline{S} the following conditional probabilities:

a. p(TH), b . p(T-H).

c. p(HIT).

Answer 1

The probability of a healthy person testing negative is 68.2%, the probability of a healthy person testing positive is 31.8%, and the probability of a healthy person testing ambiguous is 0%.

Step-by-step explanation:

The correct answer is option (a): p(TH) = 0

Conditional probability p(TH) represents the probability that the test is negative (T-) given that the person is healthy (H). According to the information given, a blood sugar level X ≤ 110 is considered normal for a healthy person, and the blood sugar measurement X for a healthy person follows a Gaussian N(90, 20) distribution.

Since 110 is within one standard deviation of the mean (90), the probability of observing a measurement X ≤ 110 for a healthy person is approximately 68.2%. Therefore, the probability of the test being negative (T-) given that the person is healthy (H), p(TH), is 68.2%.

The correct answer is option (b): p(T-H) = 31.8%

Conditional probability p(T-H) represents the probability that the test is positive (T+) given that the person is healthy (H). Since the probability of the test being negative (T-) given that the person is healthy (H) is 68.2%, the probability of the test being positive (T+) given that the person is healthy (H) is 31.8%. Therefore, p(T-H) is 31.8%.

The correct answer is option (c): p(HIT) = 0

Conditional probability p(HIT) represents the probability that the person is healthy (H) given that the test is positive (T+). Since the result is called positive (T+) when X ≥ 140, which is beyond two standard deviations of the mean (90), the probability of observing a blood sugar level X ≥ 140 for a healthy person is extremely low. Therefore, p(HIT) is 0.