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Osteoporosis. Osteoporosis is a condition in which the bones become brittle due to loss of minerals. To diagnose osteoporosis, an elaborate apparatus measures bone mineral density (BMD). BMD is usually reported in standardized form. The standardization is based on a population of healthy young adults. The World Health Organization (WHO) criterion for osteoporosis is a BMD lower than \(2.5\) standard deviations below the mean for healthy young adults. BMD measurements in a population of people similar in age and sex roughly follow a Normal distribution. a. What percentage of healthy young adults have osteoporosis by the WHO criterion? b. Women aged \(70-79\) are, of course, not young adults. The mean BMD in this age is about \(-2\) on the standard scale for young adults. Suppose the standard deviation is the same as for young adults. What percentage of this older population have osteoporosis?

Short Answer

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a. 0.62%; b. 30.85% of older women have osteoporosis.

Step by step solution

01

Understand the WHO Criterion for Osteoporosis

The WHO criterion for osteoporosis is defined as having a Bone Mineral Density (BMD) lower than 2.5 standard deviations below the mean for healthy young adults. This means that individuals with BMD values falling more than 2.5 standard deviations below the mean are considered to have osteoporosis.
02

Calculate Percentage in a Normal Distribution (Part a)

For part a, we are looking for the percentage of healthy young adults with a BMD less than 2.5 standard deviations below the mean. In a normal distribution, values below -2.5 standard deviations correspond to the lower tail. Using the standard normal distribution table (z-table), the cumulative probability for a z-score of -2.5 is approximately 0.0062, which means about 0.62% of healthy young adults fall into this category.
03

Assess BMD of Older Adults Relative to Young Adults

In part b, the mean BMD for women aged 70-79 is about -2 on the standard scale, with the same standard deviation as young adults. To find what percentage of this older group is considered to have osteoporosis, we need to find the percentage of their BMDs falling below -2.5. Since their mean is -2, we calculate the relative z-score as \(z = \frac{-2.5 - (-2)}{1} = -0.5\).
04

Calculate Percentage of Older Population with Osteoporosis (Part b)

Now, we use the z-table to find the percentage of individuals with BMD values below this z-score. A z-score of -0.5 corresponds to a cumulative probability of approximately 0.3085 or 30.85%. Thus, 30.85% of women aged 70-79 have a BMD considered osteoporosis according to the WHO criterion.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Normal Distribution
The normal distribution is a way to represent data that follows a bell-shaped curve. This curve is symmetrical, with most of the data points lumped around the mean (average). As you move away from the mean, there are fewer occurrences of data points. This distribution is quite common in naturally occurring datasets and is used extensively in statistics to make inferences about a population.

One of the key features of a normal distribution is that it is defined by its mean and standard deviation. The mean determines the center of the curve, while the standard deviation determines the spread or, how "wide" the curve is.

In a standard normal distribution, the mean is often represented as 0, and the standard deviation as 1, allowing the creation of a z-table that helps in finding probabilities for various z-scores (values along the x-axis of the normal distribution). This makes it very useful for calculations in statistics, like determining osteoporosis risk based on bone mineral density.
Standard Deviation
Standard deviation is a measure of how spread out numbers are in a data set. It tells us how much individual data points deviate from the mean of the data set. A low standard deviation means that most of the numbers are close to the mean, while a high standard deviation indicates that the numbers are more spread out.

In the context of bone mineral density (BMD), if the standard deviation is high, there would be a wide variation in BMD values among the population. In simpler terms, there would be great differences in BMD from one individual to another.

When talking about osteoporosis, the World Health Organization uses standard deviation as a part of its criteria. Specifically, a BMD value of more than 2.5 standard deviations below the average for healthy young adults, indicates osteoporosis.
World Health Organization Criterion
The World Health Organization (WHO) criterion for osteoporosis is a key diagnostic measure used worldwide. According to the WHO, an individual is considered to have osteoporosis if their bone mineral density (BMD) is more than 2.5 standard deviations below the average BMD of a healthy young adult population.

This criterion is important because it standardizes what is considered "normal" or "abnormal" BMD across different populations. Using a standard measure helps healthcare providers around the world make consistent diagnoses and develop appropriate treatment plans.

It's also worth noting that while this criterion is universally accepted, the actual risk of fractures varies between individuals and populations due to various factors including age and sex.
Osteoporosis Diagnosis
Diagnosing osteoporosis involves measuring an individual's bone mineral density (BMD) and comparing it against a standard reference. The WHO criterion is applied here to determine if an individual's BMD is significantly lower than that of a healthy young adult.

As part of the diagnosis process, health professionals typically use machines like DEXA scanners to measure BMD. The resulting value is then converted into a score known as a "T-score."
  • If the T-score is -1.0 or above, it indicates normal bone density.
  • If the T-score is between -1.0 and -2.5, it indicates osteopenia, a condition that precedes osteoporosis.
  • If the T-score is -2.5 or lower, it is indicative of osteoporosis.
Using this standardized approach not only assists in diagnosing but also in monitoring the progress of the disease and the efficacy of treatments. Understanding and using these scores are crucial steps in tackling osteoporosis effectively.

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