Glucose Level Problem

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 = 86 and standard deviation σ = 27. 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

(a)

For a probability of of getting a glucose value of 60 or more, given a normal distribution we will need to find the z value for x = 60.

z values are calculated by taking the target value x (in this case glucose reading of 60) and subtracting the mean value (μ  = 86). The result is divided by the standard deviation (σ = 27). z = (60 - 86)/27

which yields z = -26/27. z = -0.96 rounded to two decimal places.

Going to the normal distribution tables we find the that probability of hitting a value with this z or less is 0.1685 or 16.85%

But, we want to know what the probability is for greater than a glucose reading of 60.

To find that we subtract the z probability from 1 (100%).

Probability of getting a reading of 60 or higher is 1 - 0.1685 = 0.8315 or 83.15%

(b)

What is the probability of a x (glucose level) is less than 110?

Need to calculate a new z value with x = 110. z = (110 - 86)/27. Yields z = 24/27. z = 0.89

Going to the normal distribution tables we find the that probability of hitting a value with this z or less is

0.8133 or 81.33%

(c)

What is the probability that the glucose value will be between 60 and 110?

This is easier as we already have the probabilities for less than 60 and less than 110.

for x = 60 we got a probability of 0.1685 and for x = 110 got a probability of 0.8133.

Probability of being between 60 and 110 is 0.8133 - 0.1685. Which equals 0.6448 or 64.48%.

(d)

What is the probability of x is greater than 125 (borderline diabetes starts at 125)? 

Need to calculate a new z value with x = 125. z = (125 - 86)/27 yields z = 39/27. z = 1.44

Going to the normal distribution tables we find the that probability of hitting a value with this z or less is

But, we want to know what the probability is, for greater than a glucose reading of 125.

To find that we subtract the z probability from 1 (100%).

Probability of getting a reading of 125 or higher is 1 - 0.9251 = 0.0749 or 7.49%