Chapter 7.3, Problem 25P

Medical: Blood Glucose 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. After a 12-hour fast, the random variable x will have a distribution that is approximately normal with mean $\mu =85$ and standard deviation $\sigma =25$ (Source: Diagnostic Tests with Nursing Implications, edited by S. Loeb, Springhouse Press). Note: After 50 years of age. both the mean and standard deviation tend to increase.

What is the probability that, for an adult (under 50 years old) after a 12-hour fast,

(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)?

To determine

The probability that, for an adult after a 12-hour fast, x is more than 60.

Answer to Problem 25P

Solution:

The probability that x is more than 60, for an adult after a 12-hour fast is 0.8413.

Explanation of Solution

We have random variable x having a normal distribution with $\mu =85,\sigma =25$

$x=60$

By using formula for normal distribution:-

$\begin{array}{l}z=\frac{x-\mu }{\sigma }\\ z=\frac{60-85}{25}\\ z=-1.0\\ P(x>60)=P(z>-1.0)\\ P(x>60)=1-P(z\le -1.0)\end{array}$

By using Table 3 from appendix:

$\begin{array}{l}P(x>60)=1-P(z\le -1.0)\\ P(x>60)=1-0.1587\\ P(x>60)=0.8413\end{array}$

The probability that x is more than 60, for an adult after a 12-hour fast is 0.8413.

To determine

The probability that x is less than 110, for an adult after a 12-hour fast.

Answer to Problem 25P

Solution:

The probability that x is less than 110, for an adult after a 12-hour fast is 0.8413.

Explanation of Solution

We have random variable x having a normal distribution with $\mu =85,\sigma =25$

$x=110$

By using formula for normal distribution:-

$\begin{array}{l}z=\frac{x-\mu }{\sigma }\\ z=\frac{110-85}{25}\\ z=1.0\\ P(x<110)=P(z<1.0)\end{array}$

By using Table 3 from appendix:

$P(x<110)=0.8413$

The probability that x is less than 110, for an adult after a 12-hour fast is 0.8413.

To determine

The probability that x is between 60 and 110, for an adult after a 12-hour fast.

Answer to Problem 25P

Solution:

The probability that x is between 60 and 110, for an adult after a 12-hour fast is 0.6826.

Explanation of Solution

We have random variable x having a normal distribution with $\mu =85,\sigma =25$

$\begin{array}{l}{x}_1=60\\ x_2=110\end{array}$

By using formula for normal distribution:-

$\begin{array}{l}z=\frac{x-\mu }{\sigma }\\ z_1=\frac{60-85}{25}=-1.0\\ z_2=\frac{110-85}{25}=1.0\\ P(60\le x\le 110)=P(-1.0\le z\le 1.0)\\ P(60\le x\le 110)=P(z\le 1.0)-P(z\le -1.0)\end{array}$

By using Table 3 from appendix:

$\begin{array}{l}P(60\le x\le 110)=0.8413-0.1587\\ P(60\le x\le 110)=0.6826\end{array}$

The probability that x is between 60 and 110, for an adult after a 12-hour fast is 0.6826.

To determine

The probability that x is greater125, for an adult after a 12-hour fast.

Answer to Problem 25P

Solution:

The probability that x is greater 125, for an adult after a 12-hour fast is 0.0548.

Explanation of Solution

We have random variable x having a normal distribution with $\mu =85,\sigma =25$

$x=125$

By using formula for normal distribution:-

$\begin{array}{l}z=\frac{x-\mu }{\sigma }\\ z=\frac{125-85}{25}\\ z=1.60\\ P(x>125)=P(z>1.60)\\ P(x>125)=1-P(z\le 1.60)\end{array}$

By using Table 3 from appendix:

$\begin{array}{l}P(x>125)=1-0.9452\\ P(x>125)=0.0548\end{array}$

The probability that x is greater 125, for an adult after a 12-hour fast is 0.0548.