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 = 25. 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 140 (borderline diabetes starts at 140)

Answer 1

0.85083 ; 0.83147 ; 0.6823 ; 0.015386

Explanation:

Given that:

σ = 25

μ = 86

(a) x is more than 60

P(x > 60)

obtain standardized score (Zscore)

Zscore = (x - μ) / σ

Z = (60 - 86) /25

Z = - 1.04

P(Z > - 1.04) = 0.85083 (Z probability calculator)

(b) x is less than 110

P(x < 110)

obtain standardized score (Zscore)

Zscore = (x - μ) / σ

Z = (110 - 86) /25

Z = 0.96

P(Z < 0.96) = 0.83147 (Z probability calculator)

(c) x is between 60 and 110

P(x < 110) - P(x < 60)

P(Z < 0.96) - P(Z < - 1.04)

0.83147 - 0.14917

= 0.6823

(d) x is greater than 140

P(x > 140)

obtain standardized score (Zscore)

Zscore = (x - μ) / σ

Z = (140 - 86) /25

Z = 2.16

P(Z > 2.16) = 0.015386 (Z probability calculator)