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 = 90 and standard deviation σ = 20. 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 = = 90

standard deviation = = 20

(a)

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

= 1 - P((x - ) / < (60 - 90) / 20)

= 1 - P(z < -1.5)

= 1 - 0.0668

= 0.9332

P(x > 60) = 0.9332

Probability = 0.9332

(b)

P(x < 110 ) = P((x - ) / < (110 - 90) / 20)

= P(z < 1)

Using standard normal table,

P(x < 110) = 0.8413

Probability = 0.8413

(c)

P(60 < x < 110) = P((60 - 90)/ 20) < (x - ) / < (110 - 90) /20 ) )

= P(-1.5 < z < 1)

= P(z < 1) - P(z < -1.5)

= 0.8413 - 0.0668

= 0.7745

Probability = 0.7745

(d)

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

= 1 - P((x - ) / < (125 - 90) / 20)

= 1 - P(z < 1.75)

= 1 - 0.9599

= 0.0401

P(x > 125) = 0.0401

Probability = 0.