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Body Mass Index (BMI) is a simple yet widely used tool to assess whether a person falls within a healthy weight range based on their height. It is calculated using your weight in kilograms divided by the square of your height in meters:\[BMI = \frac{weight (kg)}{height (m)^2}\]Online calculators often allow you to input your weight in pounds and height in inches, automatically converting these units to meters and kilograms to perform the calculation. While BMI is a common measure, it is sometimes controversial. This is because it doesn’t distinguish between weight from fat and weight from muscle.When interpreting BMI:

• A BMI under 18.5 is considered underweight.
• A BMI between 18.5 and 24.9 is considered normal weight.
• A BMI between 25 and 29.9 suggests overweight.
• A BMI of 30 or higher is categorized as obese.

However, other factors like muscle mass, bone density, and overall body composition can affect the BMI classification. Despite these limitations, BMI is a quick, inexpensive screening tool used to highlight potential health risks.

The Z-score is a statistical measure that tells us how many standard deviations a data point is from the mean of a dataset. It is extremely helpful when working with normally distributed data, like BMI, where we want to convert individual data points into a standard context for easy comparison. The formula for finding a Z-score is:\[Z = \frac{X - \mu}{\sigma}\]Here:

• \( X \) represents the individual data point, in this case, a BMI value.
• \( \mu \) is the mean of the dataset.
• \( \sigma \) is the standard deviation.

In the given exercise, for a BMI value of 18.5, the Z-score calculation would show how far 18.5 is below the mean BMI of 26.8, with a standard deviation of 7.4. After substituting the numbers, the computation yields a Z-score of approximately \(-1.122\). This tells us that the BMI of 18.5 is 1.122 standard deviations below the average BMI of American young women.

The Standard Normal Distribution Table, commonly known as the Z-table, is a crucial tool in statistics for finding probabilities associated with a standard normal distribution. This table allows us to determine the probability that a standard normal variable (a Z-score) is less than or equal to a specific value. Here's how to use the table to find the percentile of a BMI-related Z-score: 1. **Calculate the Z-score:** As we calculated in the example, a BMI of 18.5 had a Z-score of -1.122. 2. **Use the Z-table:** Look for the value of -1.12 (since Z-tables often round to two decimal places) on the table. This gives you the area under the curve to the left of this Z-score. 3. **Interpret the table value:** In the table, a Z-score of -1.12 corresponds to a probability of approximately 0.1314. This means that about 13.14% of American young women have a BMI less than 18.5. The Z-table provides a direct pathway to relate raw scores, like specific BMI values, to their probabilities in a standard normal distribution. This conversion is critical in many statistical analyses and helps to understand where a particular measurement stands in comparison to an "average" or "mean" population.