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 = 86 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

Given,

= 86, = 26

We convert this to standard normal as

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

a)

P( X > 60) = P( Z > 60 - 86 / 26)

= P( Z > -1)

= P( Z < 1 )

= 0.8413

b)

P( X < 110) = P (Z < 110 - 86 / 26)

= P( Z < 0.9231)

= 0.8220

c)

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

= P (Z < 110 - 86 / 26) - P( Z < 60 - 86 / 26)

= P( Z < 0.9231 ) - P( Z < -1)

= 0.8220 - 0.1587

= 0.6633

d)

P( X > 125) = P( Z > 125 - 86 / 26)

= P( Z < 1.5)

= 0.9332