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 ? = 89 and standard deviation ? = 30. 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 140 (borderline diabetes starts at 140)

Answer 1

Given,

= 89, = 30

We convert this to standard normal as

P( X < x) = P( Z < x - / )

a)

P( X > 60) = P( Z > 60 - 89 / 30)

= P( Z > -0.9667)

= P( Z < 0.9667)

= 0.8331

b)

P( X < 110) = P( Z < 110 - 89 / 30)

= P( Z < 0.7)

= 0.7580

c)

P( 60 < X < 110) = P( X < 110) - P( X < 60)

= P( Z < 110 - 89 / 30) - P( X < 60 - 89 / 30)

= P( Z < 0.7) - P( Z < -0.9667)

= 0.7580 - 0.1668

= 0.5912

d)

P( X > 140) = P( Z > 140 - 89 / 30)

= P( Z > 1.7)

= 1 - P( Z < 1.7)

= 1 - 0.9554

= 0.0446