In: Statistics and Probability
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 = 81 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)
Solution :
Given that ,
mean = = 81
standard deviation = = 25
(a)
P(x > 60) = 1 - P(x < 60)
= 1 - P((x - ) / < (60 - 81) / 25)
= 1 - P(z < -0.84)
= 1 - 0.2005
= 0.7995
Probability = 0.7995
(b)
P(x < 110) = P((x - ) / < (110 - 81) / 25)
= P(z < 1.16)
= 0.877
Probability = 0.877
(c)
P(60 < x < 110) = P((60 - 81)/ 25) < (x - ) / < (110 - 81) / 25) )
= P(-0.84 < z < 1.16)
= P(z < 1.16) - P(z < -0.84)
= 0.877 - 0.2005
= 0.6765
Probability = 0.6765
(d)
P(x > 125) = 1 - P(x < 125)
= 1 - P((x - ) / < (125 - 81) / 25)
= 1 - P(z < 1.76)
= 1 - 0.9608
= 0.0392
Probability = 0.0392