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 μ = 80 and standard deviation σ = 21. 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) 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

Solution :

Given that mean μ = 80 , standard deviation σ = 21

(a) x is more than 60

=> P(x > 60) = P((x - μ)/σ > (60 - 80)/21)

= P(Z > -0.9524)

= P(Z < 0.9524)

= 0.8289

(b) x is less than 110

=> P(x < 110) = P((x - μ)/σ < (110 - 80)/21)

= P(Z < 1.4286)

= 0.9236

(c) x is between 60 and 110

=> P(60 < x < 110) = P((60 - 80)/21 < (x - μ)/σ < (110 - 80)/21)

= P(-0.9524 < Z < 1.4286)

= 0.7525

(d) x is greater than 125

=> P(x > 125) = P((x - μ)/σ > (125 - 80)/21)

= P(Z > 2.1429)

= 1 − P(Z < 2.1429)

= 1 − 0.9838

= 0.0162