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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 \(\mathrm{WHO}\) 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 WHO Criterion

WHO defines osteoporosis as a BMD score that is 2.5 standard deviations below the mean BMD for healthy young adults. In this standardized distribution of BMD, these values are represented as the negative end of the distribution.

02

Determine the Percent of Young Adults with Osteoporosis

Since the BMD follows a normal distribution, we need to find the proportion of the distribution that lies 2.5 standard deviations below the mean. In a standard normal distribution, the proportion of individuals who are more than 2.5 standard deviations below the mean is found by looking up 2.5 in a standard normal distribution table or using a calculator. This area corresponds to approximately 0.6%.

03

Analyze the Older Population (70-79 years old)

For the older population, the mean BMD is -2 (already lower than the young adult standard mean). The standard deviation remains the same.

04

Calculate the Z-Score for Osteoporosis in Older Adults

Calculate a Z-score for -2.5 standard deviations below the young adult mean in the older population. This Z-score is calculated by using the formula: \[ Z = \frac{-2.5 - (-2)}{1} = -0.5 \]. This means we need to find the cumulative probability for a Z-score of -0.5.

05

Determine the Percent of Older Adults with Osteoporosis

Using a standard normal distribution table or calculator, we find the percentage of individuals with a Z-score less than -0.5. This corresponds to the cumulative area under the curve to the left of -0.5, which is about 30.85%.

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Key Concepts

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

Osteoporosis Diagnosis

Osteoporosis is a medical condition characterized by brittle bones, mainly due to the loss of minerals like calcium. This condition can increase the risk of fractures and other bone-related issues.
Diagnosing osteoporosis typically involves measuring Bone Mineral Density (BMD). This assessment provides insight into the density and strength of bones.
The BMD values are standardized based on healthy young adults to ensure accuracy in comparison. Powerful devices known as densitometers are used in clinics and laboratories to measure BMD, providing a numerical value that helps physicians diagnose osteoporosis. Understanding BMD values in relation to established health norms is crucial in diagnosing and treating osteoporosis efficiently.

Bone Mineral Density (BMD)

Bone Mineral Density (BMD) is a measurement that reflects the amount of minerals in a segment of bone. It’s a key indicator of bone health, often measured through techniques like Dual-energy X-ray Absorptiometry (DEXA) scans.
This measurement helps determine the strength and density of bones by assessing how much calcium and other minerals they contain.
BMD results are reported in a "t-score," which compares an individual's BMD with the average BMD of a healthy young adult.

• A positive t-score means a higher bone density compared to the average young adult.
• A t-score of zero indicates average density.
• A negative t-score suggests lower density, which may indicate osteopenia or osteoporosis depending on its severity.

BMD measurement is vital in diagnosing osteoporosis, as it helps establish the risk of fractures and assess the progression of bone loss over time.

WHO Criterion

The World Health Organization (WHO) criterion provides a globally accepted standard for diagnosing osteoporosis. It defines osteoporosis as having a BMD that is 2.5 or more standard deviations below the mean BMD of a healthy, young adult population.
This means a t-score of -2.5 or lower typically indicates osteoporosis.
The WHO criterion is based on statistical principles involving normal distribution. In a normal distribution, most data points cluster around the mean, with fewer data points lying further from the mean.

• In osteoporosis assessment, only about 0.6% of healthy young adults naturally fall into the category of having osteoporosis according to this criterion, as they fall below the -2.5 threshold.
• With aging populations, however, more individuals tend to fall below this threshold due to natural bone density loss over time.

The application of this criterion helps in early diagnosis and management, as it sets a clear benchmark for when a person may need intervention or monitoring for bone health.