The body mass index (BMI) of a person is defined to be the person's body mass divided by the square of the person's height. The article "Influences of Parameter Uncertainties within the ICRP 66 Respiratory Tract Model: Particle Deposition" (W. Bolch, E. Farfan, et al„ Health Physics, 2001:378‒394) states that body mass index (in kg/m2) in men aged 25‒34 is log normally distributed with parameters μ = 3.215 and σ = 0.157.

a. Find the mean BMI for men aged 25‒34.

b. Find the standard deviation of BMI for men aged 25‒34.

c. Find the median BMI for men aged 25‒34.

d. What proportion of men aged 25‒34 have a BMI less than 22?

e. Find the 75th percentile of BMI for men aged 25‒34.

Solution 3E

Step1 of 6:

Let us consider a random variable X it presents the BMI for randomly chosen man aged 25-34.

And X follows lognormal distribution with mean and standard deviation

That is XN(3.215, 0.1572)

Here our goal is:

a).We need to find the mean BMI for men aged 25-34.

b).We need to find the standard deviation of BMI for men aged 25-34.

c).We need to find the median BMI for men aged 25-34.

d).We need to find the proportion of men aged 25.34 have a BMI less than 22.

e).We need to find the 75th percentile of BMI for men aged 25-34.

Step2 of 6:

a).

The probability density function of lognormal distribution is given by

Where,

mean

standard deviation

Z = standard normal variable.

Now, the mean BMI for men aged 25-34 is given by

=

=

=

= 25.0281

Hence, E(X) = 25.0281.

Step3 of 6:

b).

The Standard deviation of BMI for men aged 25-34 is given by

= 3.9497

Hence, =737.9516.

Step4 of 6:

c).

Here we need to find median of BMI for men aged 25-34. Let us consider “m” be the median lifetime of an electric component. And we know that median divides the data into two parts that is 0.5.

Now, median of BMI for men aged 25-34 is given by

We know that X N(3.215, 0.1572)

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