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

(a)

= 82

= 27

To find P(X>60):

Z = (60 - 82)/27

= - 0.8148

Table of Area Under Standard Normal Curve gives area = 0.2910

So,

P(X>60) = 0.5 + 0.2910 = 0.7910

So,

Answer is:

0.7910

(b)

To find P ( X < 110):

Z = (110 - 82)/27

= 1.0370

Table of Area Under Standard Normal Curve gives area = 0.3508

So,

P(X<110) = 0.5 + 0.3508 = 0.8508

So,

Answer is:

0.8508

(c)

To find P(60 < X < 110):

Case 1: For X from 60 to mid value:

Z = (60 - 82)/27

= - 0.8148

Table of Area Under Standard Normal Curve gives area = 0.2910

Case 2: For X from mid value to 110:

Z = (110 - 82)/27

= 1.0370

Table of Area Under Standard Normal Curve gives area = 0.3508

So,

P(60 < X < 110) = 0.2910 + 0.3508 = 0.6418

So,

Answer is:

0.6418

(d)

To find P(X > 125):

Z = (125 - 82)/27

= 1.5926

Table of Area Under Standard Normal Curve gives area = 0.4441

So,

P(X>125) = 0.5 - 0.4441 = 0.0559

So,

Answer is:

0.0559