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3.34 Body Mass Index. Your body mass index (BMI) is your weight in kilograms divided by the square of your height in meters. Many online BMI calculators allow you to enter weight in pounds and height in inches. High BMI is a common but controversial indicator of overweight or obesity. A study by the National Center for Health Statistics found that the BMI of 2-year-old American male children is approximately Normally distributed, with mean \(16.8\) and standard deviation 1.9.12 a. What percentage of 2 -year-old American male children have a BMI less than 15.0? b. What percentage of 2-year-old American male children have a BMI less than \(18.5\) ? Miles per Gallon. In its Fuel Economy Guide for 2019 model vehicles, the Environmental Protection Agency gives data on 1259 vehicles. There are a number of high outliers, mainly hybrid gas-electric vehicles. If we ignore the vehicles identified as outliers, however, the combined city and highway gas mileage of the other 1231 vehicles is approximately Normal with mean \(22.8\) miles per gallon (mpg) and standard deviation \(4.8 \mathrm{mpg}\). Exercises. \(3.35\) through 3.38 concern this distribution.

Step by step solution

01

Understanding the Problem

We need to find the percentage of 2-year-old American male children who have a BMI less than 15.0 and less than 18.5, given that the BMI follows a Normal distribution with mean 16.8 and standard deviation 1.9.

02

Calculate Z-scores for BMI

To calculate the percentage, we first find the Z-score for each BMI value. The Z-score formula is \( Z = \frac{X - \mu}{\sigma} \), where \( X \) is the value of interest, \( \mu \) is the mean, and \( \sigma \) is the standard deviation.For BMI less than 15.0:\[ Z = \frac{15.0 - 16.8}{1.9} \approx -0.947 \]For BMI less than 18.5:\[ Z = \frac{18.5 - 16.8}{1.9} \approx 0.895 \]

03

Find the Cumulative Probability

Using the Z-table, we find the cumulative probability, which gives the percentage of values below a given Z-score.For \( Z \approx -0.947 \), the cumulative probability is approximately 0.172.For \( Z \approx 0.895 \), the cumulative probability is approximately 0.814.

04

Convert Cumulative Probability to Percentage

To find the percentage, multiply the cumulative probability by 100.For BMI less than 15.0: \( 0.172 \times 100 \approx 17.2\% \)For BMI less than 18.5: \( 0.814 \times 100 \approx 81.4\% \)

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Key Concepts

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

Body Mass Index

The Body Mass Index (BMI) is a numerical computation that assesses the relationship between an individual's weight and height. Calculating BMI involves dividing an individual's weight in kilograms by the square of their height in meters: \[ \text{BMI} = \frac{\text{weight in kg}}{\text{height in m}^2} \]For instance, a person weighing 70 kg and standing 1.75 m tall would have a BMI of \[ \frac{70}{1.75^2} \approx 22.86 \]. BMIs are often used to categorize individuals based on weight, indicating ranges such as underweight, normal weight, overweight, and obesity. Many find it beneficial due to its ease of use; however, it is important to remember that BMI does not measure body fat directly or account for certain factors like muscle mass and distribution of weight.
When applied to populations, as in the case of 2-year-old American male children, the BMI distribution can provide insights into health trends and concerns like obesity.

Standard Deviation

Standard deviation is a statistical measurement that offers insights into the dispersion or variation in a set of values. In simpler terms, it tells us how much the numbers in a data set tend to deviate from the mean (average). A lower standard deviation indicates data points are close to the mean, while a higher value suggests a wide range of numbers. In the BMI study for 2-year-old American male children, the standard deviation is noted as 1.9. This implies that the BMI values differ by approximately 1.9 units from the average BMI of 16.8. A small standard deviation here might suggest that most children's BMI values are close to the average, whereas a larger deviation would indicate a greater spread. This is crucial in understanding whether the majority of children fall into a typical range for BMI or if there are significant differences among them. By using standard deviation, researchers can get a clearer picture of health metrics in a population.

Z-scores

Z-scores are a valuable tool in statistics to understand how far or how close a data point is from the mean in terms of standard deviations. Mathematically, the Z-score is calculated by the formula:\[ Z = \frac{X - \mu}{\sigma} \]where:

• \(X\) is the value in question,
• \(\mu\) is the mean, and
• \(\sigma\) is the standard deviation.

Z-scores transform data into a standard form, particularly useful when comparing data points from different sets or when dealing with normal distributions.In the original problem, the BMI values are converted to Z-scores to determine the relative position of a BMI of 15.0 and 18.5 against the average. For example, a Z-score of approximately -0.947 for a BMI of 15.0 indicates it is nearly 0.947 standard deviations below the mean, highlighting how rare or common this value is within the distribution.

Cumulative Probability

Cumulative probability is a concept used in statistics to determine the likelihood of a variable falling within a particular range. It represents the sum of probabilities for all values up to a specific point. Typically, a Z-table is used to find the cumulative probability associated with a Z-score. In our case of BMI, cumulative probabilities identify what fraction of 2-year-olds have a BMI below a certain figure. From the Z-scores calculated for BMIs of 15.0 and 18.5, the corresponding cumulative probabilities are found to be approximately 0.172 and 0.814, respectively. To convert these probabilities to percentages, multiply by 100. Thus, 17.2% and 81.4% of children have BMIs less than 15.0 and 18.5, respectively. This information helps public health professionals understand distribution patterns and address health conditions within a population from a broader perspective.