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 = 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 125 (borderline diabetes starts at 125)

Answer 1

Solution :

Given that ,

mean = = 89

standard deviation = = 25

(a)

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

= 1 - P[(x - ) / < (60 - 89) / 25)

= 1 - P(z < -1.16)

= 1 - 0.123

= 0.8770

Probability = 0.8770

(b)

P(x < 110) = P[(x - ) / < (110 - 89) / 25]

= P(z < 0.84)

= 0.7995

Probability = 0.7995

(c)

P(60 < x < 110) = P[(60 - 89)/ 25) < (x - ) /  < (110 - 89) / 25) ]

= P(-1.16 < z < 0.84)

= P(z < 0.84) - P(z < -1.16)

= 0.7995 - 0.123

= 0.6765

Probability = 0.6765

(d)

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

= 1 - P[(x - ) / < (125 - 89) / 25)

= 1 - P(z < 1.44)

= 1 - 0.9251

= 0.0749

Probability = 0.0749