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 = 81 and standard deviation σ = 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) 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

(a)

= 81

= 26

To find P(X>60):

Z = (60 - 81)/26

= - 0.8077

By Technology, Cumulative Area Under Standard Normal Curve = 0.2096

So,

P(X>60) = 1 - 0.2096 = 0.7904

So,

Answer is:

0.7904

(b)

= 81

= 26

To find P(X<110):

Z = (110 - 81)/26

= 1.1154

By Technology, Cumulative Area Under Standard Normal Curve = 0.8677

So,

P(X<110) = 0.8677

So,

Answer is:

0.8677

(c)

= 81

= 26

To find P(60<X<110):

For X = 60:

Z = (60 - 81)/26

= - 0.8077

By Technology, Cumulative Area Under Standard Normal Curve = 0.2096

For X = 110:

Z = (110 - 81)/26

= 1.1154

By Technology, Cumulative Area Under Standard Normal Curve = 0.8677

So,

P(60<X<110) = 0.8677 - 0.2096 = 0.6581

So,

Answer is:

0.6581

(d)

= 81

= 26

To find P(X>125):

Z = (125 - 81)/26

= 1.6923

By Technology, Cumulative Area Under Standard Normal Curve = 0.9547

So,

P(X>125) = 1 - 0.9547 = 0.0453

So,

Answer is:

0.0453