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 = 83 and standard deviation σ = 24. 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 =   = 83

standard deviation = = 24

n = 12

= 83

=  ( /n) = (24 / 12 ) = 9.9282

a ) P (   > 60)

= 1 - P (   < 60 )

= 1 -  P ( - /) < (60 - 83 / 9.9282)

= 1 - P ( z < - 23 / 9.9282 )

= 1 -  P ( z < -2.32 )

Using z table

= 1 - 0.0102

= 0.9898

Probability = 0.9898,

b ) P (   < 110)

P ( - /) < (110 - 83 / 9.9282)

P ( z < 27 / 9.9282 )

P ( z < 2.72 )

Using z table

= 0.9967

Probability = 0.9967

c ) P ( 60 < < 110)

P (60 - 83 / 9.9282) ( - /) < (110 - 83 / 9.9282)

P ( - 23 / 9.9282 < z < 27 / 9.9282 )

P ( - 2.32 < z < 2.72 )

P ( Z < 2.72 ) - P ( Z < - 2.32 )

Using z table

=0.9967 - 0.0102

= 0.9865

Probability = 0.9865

d ) P (   > 125 )

= 1 - P (   < 125 )

= 1 -  P ( - /) < (125 - 83 / 9.9282)

= 1 -  P ( z < 42 / 9.9282 )

  = 1 -  P ( z < 4.23 )

Using z table

= 1 - 1

= 0

Probability = 0