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. Suppose that after a 12-hour fast, the random variable x will have a distribution that is approximately normal with mean = 87 and standard deviation = 26. Note: After 50 years of age, both the mean and standard deviation tend to increase. For an adult (under 50) after a 12-hour fast, find the following probabilities. (Round your answers to four decimal places.)

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. Suppose that after a 12-hour fast, the random variable x will have a distribution that is approximately normal with mean = 87 and standard deviation = 26. Note: After 50 years of age, both the mean and standard deviation tend to increase. For an adult (under 50) after a 12-hour fast, find the following probabilities. (Round your answers to four decimal places.)

A button hyperlink to the SALT program that reads: Use SALT.
(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

To find the probabilities, we need to standardize the values of x using the formula:

z = (x - mean) / standard deviation

(a) To find the probability that x is more than 60, we need to find P(x > 60). Let's standardize 60:

z = (60 - 87) / 26
z ≈ -1.0385

Using the standard normal distribution table or a calculator, we can find the probability that z is less than -1.0385:

P(z < -1.0385) ≈ 0.1492

Therefore, the probability that x is more than 60 is 1 - 0.1492 = 0.8508.

(b) To find the probability that x is less than 110, we need to find P(x < 110). Let's standardize 110:

z = (110 - 87) / 26
z ≈ 0.8846

Using the standard normal distribution table or a calculator, we can find the probability that z is less than 0.8846:

P(z < 0.8846) ≈ 0.8106

Therefore, the probability that x is less than 110 is 0.8106.

(c) To find the probability that x is between 60 and 110, we need to find P(60 < x < 110). Let's standardize both values:

For 60:
z1 = (60 - 87) / 26
z1 ≈ -1.0385

For 110:
z2 = (110 - 87) / 26
z2 ≈ 0.8846

Using the standard normal distribution table or a calculator, we can find the probabilities of z being less than -1.0385 and 0.8846, and then subtract the smaller probability from the larger:

P(-1.0385 < z < 0.8846) ≈ 0.8106 - 0.1492 = 0.6614

Therefore, the probability that x is between 60 and 110 is 0.6614.

(d) To find the probability that x is greater than 125, we need to find P(x > 125). Let's standardize 125:

z = (125 - 87) / 26
z ≈ 1.4615

Using the standard normal distribution table or a calculator, we can find the probability that z is greater than 1.4615:

P(z > 1.4615) ≈ 0.0718

Therefore, the probability that x is greater than 125 is 0.0718.