Question

A person's blood glucose level and diabetes are closely related. Let x be a random variable measured in milligrams of glucose per deciliter (1/10 of a liter) of blood. After a 12-hour fast, the random variable x will have a distribution that is approximately normal with mean m=85 and standard deviation s=25 (Source: Diagnostic Tests with Nursing Implications, edited by S. Loeb, Springhouse Press). Note: After 50 years of age, both the mean and standard deviation tend to increase. What is the probability that, for an adult (under 50 years old) after a 12-hour fast, (a)x is more than 60? (b)x is less than 110? (c)x is between 60 and 110? (d)x is greater than 125 (borderline diabetes starts at 125)?

Answer 1

The probability that a person's blood glucose level is more than 60 after a 12-hour fast is approximately 0.8413. The probability that a person's blood glucose level is less than 110 is approximately 0.8413. The probability that a person's blood glucose level is between 60 and 110 is approximately 0.6826.

Step-by-step explanation:

To find the probability that a person's blood glucose level is more than 60 after a 12-hour fast, we need to calculate the z-score and use a standard normal distribution table or calculator. The z-score formula is given by z = (x - µ)/σ, where x is the value of the random variable, µ is the mean, and σ is the standard deviation.

So, for (a) x > 60:
z = (60 - 85) / 25 = -1

Using a z-score table or calculator, we find that the probability of z < -1 is approximately 0.1587. Since we want the probability of x > 60, we subtract this value from 1:
P(x > 60) ≈ 1 - 0.1587 = 0.8413

Similarly, for (b) x < 110:
z = (110 - 85) / 25 = 1
Using a z-score table or calculator, we find that the probability of z < 1 is approximately 0.8413.
P(x < 110) ≈ 0.8413

For (c) 60 < x < 110:
First, find the probability of x < 60:
z = (60 - 85) / 25 = -1
P(x < 60) = 0.1587
Then subtract this from the probability of x < 110:
P(60 < x < 110) ≈ 0.8413 - 0.1587 = 0.6826

Lastly, for (d) x > 125:
z = (125 - 85) / 25 = 1.6
Using a z-score table or calculator, we find that the probability of z < 1.6 is approximately 0.9452.
P(x > 125) ≈ 1 - 0.9452 = 0.0548