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 = 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

Solution :

(a)

P(x > 60) = 1 - P(x < 60)

= 1 - P[(x - ) / < (60 - 82) / 21)

= 1 - P(z < -1.05)

= 1 - 0.1469

= 0.8531

Probability = 0.8531

(b)

P(x < 110) = P[(x - ) / < (110 - 82) / 21]

= P(z < 1.33)

= 0.9082

Probability = 0.9082

(c)

P(60 < x < 110) = P[(60 - 82)/ 21) < (x - ) /  < (110 - 82) / 21) ]

= P(-1.05 < z < 1.33)

= P(z < 1.33) - P(z < -1.05)

= 0.9082 - 0.1469

= 0.7613

Probability = 0.7613

(d)

P(x > 125) = 1 - P(x < 125)

= 1 - P[(x - ) / < (125 - 82) / 21)

= 1 - P(z < 2.05)

= 1 - 0.9798

= 0.0202

Probability = 0.0202