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 = 22. 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)


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Answer 1

You require a statistical calculator or a normal probability table to evaluate the probabilities.

A)
μ = 87
σ = 27
standardize x to z = (x - μ) / σ
P(x > 60) = P( z > (60-87) / 27)
= P(z > -1) = 0.8413
(From Normal probability table)

B)
μ = 87
σ = 27
standardize x to z = (x - μ) / σ
P(x < 110) = P( z < (110-87) / 27)
= P(z < 0.8519) = 0.8023
(From Normal probability table)

C)
μ = 87
σ = 27
standardize x to z = (x - μ) / σ
P( 60 < x < 110) = P[( 60 - 87) / 27 < Z < ( 110 - 87) / 27]
P( -1 < Z < 0.8519) = 0.3413+0.3023 =0.6436
(From Normal probability table)

D)
μ = 87
σ = 27
standardize x to z = (x - μ) / σ
P(x > 140) = P( z > (140-87) / 27)
= P(z > 1.963) = 0.025
(From Normal probability table)