Problem 19 - Algebra 1

Question

Body mass index is a measure of weight in relation to height. The table shows the body mass index \(B\) of a person who is 152 centimeters tall and weighs \(w\) kilograms. Which type of model best fits the data? Write a model. $$\begin{array}{|l|c|c|c|c|c|c|c|}\hline \text { Weight, } w & 45.40 & 49.94 & 54.48 & 59.02 & 63.56 & 68.10 & 72.64 \\\\\hline \text { Body mass index, } B & 19.55 & 21.50 & 23.46 & 25.41 & 27.37 & 29.32 & 31.28 \\\\\hline\end{array}$$

Step-by-Step Solution

Answer

The relationship between the weight and the body mass index suggests a linear model. This model can be represented as \(y = mx + c\), where \(y\) is the body mass index, \(x\) is the weight, \(m\) is the slope, and \(c\) is the y-intercept. The exact values of \(m\) and \(c\) can be calculated using linear regression analysis.

Step 1: Identify the type of model

By analyzing the data in the table, it can be observed that for every increase in weight, there's a corresponding increase in body mass index. This proportional change suggests a linear relationship between the two variables. Thus, a linear model can represent this relationship.

Step 2: Perform linear regression analysis

To confirm this hypothesis and find the exact linear model, perform linear regression analysis. The linear model will be in the form of \(y = mx + c\), where \(y\) is the body mass index, \(x\) is the weight, \(m\) is the slope (the change in \(y\) for each unit change in \(x\)), and \(c\) is the \(y\)-intercept (the value of \(y\) when \(x\) equals 0). The linear regression analysis will give the values of \(m\) and \(c\).

Step 3: Write the model

Once \(m\) and \(c\) have been calculated from the linear regression analysis, substitute these values into the linear model form \(y = mx + c\). This will yield the model that describes the relationship between the body weight and body mass index. Note that this step requires computational analysis or the use of software to compute the exact values of \(m\) and \(c\).

Key Concepts

Understanding Body Mass Index

Body mass index (BMI) is a widely used metric to assess whether an individual has a healthy body weight relative to their height. It's calculated by taking a person's weight in kilograms and dividing it by the square of their height in meters (\( BMI = \frac{weight(kg)}{height(m)^2} \)).

This calculation provides a simple numerical measure that can categorize individuals as underweight, normal weight, overweight, or obese, based on defined ranges. For example, a BMI below 18.5 is considered underweight, while a BMI over 30 indicates obesity. However, it's important to note that BMI is not a perfect measure as it does not differentiate between weight from muscle and weight from fat, and it does not take into account the distribution of fat throughout the body.

The Essence of Linear Relationships

In statistics and mathematics, a linear relationship between two variables is one where the rate of change along one variable is constant relative to the other. This means that a straight line graphically represents the relationship between them on a scatter plot. When we increase or decrease one variable, the other increases or decreases at a consistent rate. In the context of BMI, when we examine data that suggest a proportional increase of BMI with weight, it indicates a linear relationship.

Understanding the nature of such relationships is crucial because it enables us to predict values and create models that can be applied to various fields, such as economics, biological sciences, and engineering. When we know that two quantities are linearly related, we can use this information to predict or control outcomes.

Modeling with Linear Equations

Modeling with linear equations is a fundamental aspect of data analysis that allows us to understand and predict behaviors within systems. In the exercise, the relationship between body weight and BMI suggests the use of a linear equation, typically written as \( y = mx + c \), where \( y \) represents the dependent variable (BMI, in this case), \( x \) the independent variable (weight), \( m \) the slope of the line (indicating how much \( y \) changes for a unit change in \( x \) ), and \( c \) the y-intercept (the value of \( y \) when \( x \) is zero).

By performing linear regression analysis on the given data, we can find the most accurate values of \( m \) and \( c \) to model this relationship. Through the process, we use statistical methods to minimize the distance between data points and the line of best fit, ensuring our model closely represents the observed data. The final equation can then be used to predict BMI for a given weight or vice versa, which is invaluable in healthcare, fitness planning, and related fields.