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Dem bones \((2.2)\) 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 score that is 2.5 standard deviations below the mean for young adults. BMD measurements in a population of people similar in age and gender roughly follow a Normal distribution. (a) What percent of healthy young adults have osteoporosis by the WHO criterion? (b) Women aged 70 to 79 are, of course, not young adults. The mean BMD in this age group is about \(-2\) on the standard scale for young adults. Suppose that the standard deviation is the same as for young adults. What percent of this older population has osteoporosis?

Step by step solution

01

Understand the Criterion

The WHO criterion for osteoporosis is a BMD score that is 2.5 standard deviations below the mean of healthy young adults. Knowing this criterion helps us determine the percentage of people who fall below this threshold in the normal distribution of BMD scores.

02

Determine the Z-score and Percentage for Young Adults

For healthy young adults, a BMD score that is 2.5 standard deviations below the mean corresponds to a Z-score of \(-2.5\). In a standard normal distribution, we look for the cumulative probability of Z ≤ -2.5. Using a Z-table or a statistical calculator, we find that approximately 0.62% of the population has a BMD score this low. This means 0.62% of healthy young adults meet the osteoporosis criterion.

03

Contextualize for Age Group 70-79

Women aged 70 to 79 have a mean BMD score of -2. To find the percentage with osteoporosis, we need the Z-score. These women have a mean Z-score equivalent of -2, and considering the criteria for osteoporosis is having a score of -2.5, we need to find the cumulative probability for Z ≤ -0.5 (as their mean is already -2).

04

Calculate the Percentage for Older Population

Using the standard normal distribution, find the cumulative probability for Z ≤ -0.5. Referring to a Z-table or calculator, we find that roughly 30.85% of the population has a Z-score less than -0.5. Therefore, about 30.85% of women aged 70 to 79 meet the osteoporosis criterion according to this analysis.

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Key Concepts

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

Osteoporosis

Osteoporosis is a medical condition where bones become fragile and are more likely to fracture. This happens because bones lose minerals like calcium. Osteoporosis is a common condition, especially in older adults. The risk of developing this disease increases with age, and women are more likely than men to develop it.
To diagnose osteoporosis, doctors use a special measure called Bone Mineral Density (BMD), which helps them understand how strong or weak your bones are. If your BMD is much lower than that of a healthy young adult, this indicates a higher risk of fractures and may lead to a diagnosis of osteoporosis. Understanding your BMD score can help you and your healthcare provider make better decisions about treatment and lifestyle changes to strengthen your bones.

Bone Mineral Density

Bone Mineral Density (BMD) is a term used to describe the amount of minerals, particularly calcium, in your bones. BMD gives us an idea of how strong and dense bones are. A higher BMD means stronger bones, while a lower BMD indicates weaker bones that are more prone to fractures.
BMD is measured using a bone densitometry test, often called a DXA scan. The results are compared to the average peak bone density of a healthy young adult, and the score is adjusted based on age, gender, and other factors. This process helps in diagnosing conditions like osteoporosis, where bone density significantly decreases.
Regular BMD testing is important, especially for older adults and postmenopausal women, as it provides crucial information on bone health and potential fracture risk.

Normal Distribution

In statistics, a normal distribution is a bell-shaped curve that shows how data is spread out. Most of the time, the data values cluster around the mean, and the probability of occurrence of values decreases as they move away from the mean.
For Bone Mineral Density (BMD), the normal distribution helps predict the likelihood of various BMD scores in a population. For instance, if young adults' BMD follows a normal distribution, we can use statistical methods to determine how many people have extremely low or high BMD scores.

• The mean represents the average BMD score.
• The standard deviation shows how tightly data points cluster around the mean.

Knowing the normal distribution allows healthcare professionals to understand better which BMD scores are typical or atypical.

Z-score

A Z-score is a statistical measure that tells us how far a particular value is from the mean of a set of values, in terms of standard deviations. When it comes to Bone Mineral Density (BMD), a Z-score helps us understand where a particular BMD measurement falls in relation to a normal distribution of BMD scores.
The formula for calculating a Z-score is:\[ Z = \frac{(X - \mu)}{\sigma} \]Where:

• \(X\) is the actual BMD score.
• \(\mu\) is the mean of the BMD for a reference population.
• \(\sigma\) is the standard deviation of the BMD scores.

A negative Z-score means the BMD is below the mean, while a positive Z-score means it is above the mean. For osteoporosis diagnosis, a Z-score of \(-2.5\) or lower indicates a diagnosis of osteoporosis according to WHO guidelines.