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 = 82 and standard deviation σ = 27. 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

x:  glucose per deciliter (1/10 of a liter) of blood

x approximately normal with mean = 82 and standard deviation = 27

(a) probability that x is more than 60 = P(x>60)

P(x>60) = 1-P(x<60)

z-score for 60 = (60-82)/27 = -0.81

From standard normal tables, P(z<-0.81) = 0.2090

P(x<60)=P(z<-0.81) = 0.2090

P(x>60) = 1-P(x<60)=1-0.2090=0.791

probability that x is more than 60 = 0.7910

(b) probability that x is less than 110 = P(x<110)

z-score for 100= (110-82)/27 =1.04

From standard normal tables, P(z<1.04) = 0.8508

P(x<110) = P(z<1.04) = 0.8508

probability that x is less than 110 = 0.8508

(c) probability that x is between 60 and 110 = P(60<x<110)=P(x<110)-P(x<60)

From (b) P(x<110) = 0.8508

From (a) P(x<60) = 0.2090

P(60<x<110)=P(x<110)-P(x<60) = 0.8508-0.2090= 0.6418

probability that x is between 60 and 110 = 0.6418

(d) probability that x is more than 125= P(x>125)=1-P(x<125)

z-score for 125 =(125-82)/27=1.59

From standard normal tables , P(z<1.59) = 0.9441

P(x<125) = P(z<1.59) = 0.9441

P(x>125)=1-P(x<125) =1-0.9441=0.0559

probability that x is more than 125=  0.0559