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 = 89 and standard deviation σ = 21. 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

x be a random variable measured in milligrams of glucose per deciliter (1/10 of a liter) of blood after a 12-hour fast

mean, μ = 89

standard deviation σ = 21

(a) x is more than 60 = P[ X > 60 ]

P[ X > 60 ] = P[ ( X - μ )/σ > ( 60 - μ )/σ ] = P[ ( X - 89 )/21 > ( 60 - 89 )/21 ]

P[ X > 60 ] = P[ Z > -1.38 ] = 1 - P[ Z < -1.38 ] = 1 - 0.0838 = 0.9162

P[ X > 60 ] = 0.9162

(b) x is less than 110 = P[ X < 110 ]

P[ X < 110 ] = P[ ( X - μ )/σ < ( 110 - μ )/σ ] = P[ ( X - 89 )/21 < ( 110 - 89 )/21 ]

P[ X < 110 ] = P[ Z < 1 ] = 0.8413

P[ X < 110 ] = 0.8413

(c) x is between 60 and 110 = P[ 60 < X < 110 ]

P[ 60 < X < 110 ] = P[ X < 110 ] - P[ X < 60 ]

P[ X < 60 ] = 1 - P[ X > 60 ] = 1 - 0.9162 = 0.0838

P[ 60 < X < 110 ] = 0.8413 - 0.0838 = 0.7575

P[ 60 < X < 110 ] = 0.7575

(d) x is greater than 125= P[ X > 125 ]

P[ X > 125 ] = P[ ( X - μ )/σ > ( 125 - μ )/σ ] = P[ ( X - 89 )/21 > ( 125 - 89 )/21 ]

P[ X > 125 ] = P[ Z > 1.71 ] = 1 - P[ Z < 1.71 ] = 1 - 0.9564 = 0.0436

P[ X > 125 ] = 0.0436