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 ? = 82 and standard deviation ? = 20. 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

Given, = 82, = 20

We convert this to standard normal as

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

a)

P( X > 60) = P( Z > 60 - 82 / 20)

= P( Z < -1.1)

= P( Z < 1.1)

= 0.8643

b)

P( X < 110) = P( Z < 110 - 82 / 20)

= P( Z < 1.4)

= 0.9192

c)

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

= P( Z < 110 - 82 / 20) - P( Z < 60 - 82 / 20)

= P( Z < 1.4) - P( Z < -1.1)

= P( Z < 1.4) - ( 1 - P( Z < 1.1) )

= 0.9192 - ( 1 - 0.8643)

= 0.7836

d)

P( X > 125) = P( Z > 125 - 82 / 20)

= P( Z > 2.15)

= 1 - P( Z < 2.15)

= 1 - 0.9842

= 0.0158