Question

BMI (2.2, 5.2, 5.3) Your body mass index (BMI) is your weight in kilograms divided by

the square of your height in meters. Online BMI calculators allow you to enter weight in

pounds and height in inches. High BMI is a common but controversial indicator of being

overweight or obese. A study by the National Center for Health Statistics found that the

BMI of American young women (ages 20 to 29) is approximately Normally distributed

with mean 26.8 and standard deviation 7.4.

27

a. People with BMI less than 18.5 are often classed as “underweight.” What percent of

young women are underweight by this criterion?

b. Suppose we select two American young women in this age group at random. Find the

probability that at least one of them is classified as underweight.

Short answer

a.percent of young women that are underweight by this criterion is $13.101$

b,.the probability that at least one of them is classifiied as underweight is$0.244$

Step-by-step solution

Part (a) Step 1:Given Information

We have been given

mean ($\mu$)=$26.8$

standard deviation($\sigma$) =$7.4$

People with BMI less than$18.5$ are often classed as “underweight.”

Part (a) Step 2:Simplification

For normal distribution we will calculate z value

z=$\frac{\chi -\mu }{\sigma }$

here,x=$18.5$

so,z=$\frac{18.5-26.8}{7.4}$

we need to find p($\chi <18.5$) as this will be considered as underweight

so,p($\chi <18.5$)=p(z<$\frac{\chi -\mu }{\sigma }$)

p($\chi <18.5$)=p(z<$\frac{18.5-26.8}{7.4}$)

p($\chi <18.5$)=p(z<$-1.121$)

p($\chi <18.5$)=0.1310

percent of young women that are underweight by this criterion is $0.1310*100$=$13.10$

Part (b) Step 1:Given Information 

Number of american young women selected randomly (n) =$2$

Part (b) Step 2:Simplification

probability that a person is underweight is (p) = $0.1310$

probability that a person is not underweight is (q) = $1-$p = $1-0.1310$=$0.869$

probability that at least one of them is classified as underweight is :p($\chi \ge 1$)=$1-$p($\chi <1$)=$1-$p($0$)

p($\chi \ge 1$)=$1-0.{869}^2$

p($\chi \ge 1$)=$0.244$