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 = 88 and standard deviation σ = 28. 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 = = 88

standard deviation = = 28

a)

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

=1 - P((x - ) / < (60 - 88) / 28)

=1- P(z < -1)

= 1 - 0.1587 Using standard normal table,

Probability = 0.8413

b)

P(x < 110) = P((x - ) / < (110 - 88) / 28)

= P(z < 0.79)

= 0.7852 Using standard normal table,

Probability = 0.7852

c)

P(60 < x < 110) = P((60 - 88)/ 28) < (x - ) /  < (110 - 88) / 28) )

= P(-1 < z < 0.79)

= P(z < 0.79) - P(z < -1)

= 0.7852 - 0.1587 Using standard normal table,  

Probability = 0.6265

d)

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

=1 - P((x - ) / < (125 - 88) / 28)

=1- P(z < 1.32)

= 1 - 0.9066 Using standard normal table,

Probability = 0.0934