Chapter 7.1, Problem 31E

(a)

To determine

To Calculate: the percent of healthy young adults have osteoporosis by the WHO criterion.

Answer to Problem 31E

0.62%

Explanation of Solution

Given:

The WHO criterion is a BMD score that is 2.5 and the standard deviations below the mean.

The population distribution is approximately normal.

Calculation:

2.5 standard deviations below the mean is equivalent with the z-score being equal to -2.5.

  $z=-2.5$

The corresponding probability using the normal probability

  $P\left(z<-2.5\right)=0.0062=0.62\%$

Therefore 0.62% of the healthy young adults have osteoporosis by the WHO criterion.

(b)

To determine

To Calculate: the percent of this older population has osteoporosis.

Answer to Problem 31E

30.85%

Explanation of Solution

Given:

Population is a normal distribution with:

  $\begin{array}{l}\mu =-2\\ \sigma =1\end{array}$

Calculation:

The z-score is

  $\begin{array}{c}z=\frac{x-\mu }{\sigma }\\ =\frac{-2.5-\left(-2\right)}{1}\\ =-0.5\end{array}$

The corresponding probability

  $P\left(X<-2.5\right)=P\left(Z<-0.5\right)=0.3085=30.85\%$

Therefore $30.85\%$ of women aged 70 to 79 have osteoporosis.