A persons blood glucose level and diabetes are closely relat

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 = 83 and standard deviation = 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 140 (borderline diabetes starts at 140)

Solution

Normal Distribution
Mean ( u ) =83
Standard Deviation ( sd )=25
Normal Distribution = Z= X- u / sd ~ N(0,1)
a)
P(X > 60) = (60-83)/25
= -23/25 = -0.92
= P ( Z >-0.92) From Standard Normal Table
= 0.8212
b)
P(X < 110) = (110-83)/25
= 27/25= 1.08
= P ( Z <1.08) From Standard Normal Table
= 0.8599                  
c)              
To find P(a < = Z < = b) = F(b) - F(a)
P(X < 60) = (60-83)/25
= -23/25 = -0.92
= P ( Z <-0.92) From Standard Normal Table
= 0.17879
P(X < 110) = (110-83)/25
= 27/25 = 1.08
= P ( Z <1.08) From Standard Normal Table
= 0.85993
P(60 < X < 110) = 0.85993-0.17879 = 0.6811                  
d)
P(X > 140) = (140-83)/25
= 57/25 = 2.28
= P ( Z >2.28) From Standard Normal Table
= 0.0113