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 = 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

0.7673 (correct answer)

(b) x is less than 110

0.8238 (correct answer)

(c) x is between 60 and 110

0.0565 (incorrect answer)

**This is the part I am struggling with

(d) x is greater than 125 (borderline diabetes starts at 125)

0.0764 (correct answer)

Answer 1

Solution :

(a)

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

= 1 - P[(x - ) / < (60 - 82) / 30]

= 1 - P(z < -0.73)

= 0.7673

(b)

P(x < 110) = P[(x - ) / < (110 - 82) / 30]

= P(z < 0.93)

= 0.8238

(c)

P(60 < x < 110) = P[(60 - 82) / 30) < (x - ) /  < (110 - 82) / 30) ]

= P(-0.73 < z < 0.93)

= P(z < 0.93) - P(z < -0.73)

= 0.8238 - 0.2327

= 0.5911

(d)

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

= 1 - P[(x - ) / < (125 - 82) / 30]

= 1 - P(z < 1.43)

= 0.0764