Problem 46 - Calculus an Applied Approach

Question

Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. Data that are modeled by \(y=-0.238 x+25\) have a negative correlation.

Step-by-Step Solution

Answer

The statement is true. Data modeled by \(y=-0.238 x+25\) indeed represent a negative correlation as the slope of the model or regression line is negative (-0.238), suggesting that as 'x' increases, 'y' decreases.

Step 1: Understand Correlation and its Representation in Linear Model

Correlation between two variables can be positive, zero or negative. A positive correlation signifies that as one variable increases, the other also increases. A zero correlation signifies no direct linear relationship between two variables. And a negative correlation signifies that as one increases, the other decreases. In a linear model \(y= ax + b\), 'a' is called the slope. If 'a' is positive, there is a positive correlation, if 'a' is zero, there is no correlation, and if 'a' is negative, there is a negative correlation.

Step 2: Analyzing the Provided Model

The provided model is \(y=-0.238 x+25\). The slope here is -0.238 which is negative.

Step 3: Confirm the Statement

As 'a' (the slope) is negative, the correlation for data modeled by the relationship \(y=-0.238 x+25\) is indeed negative. Hence, the statement is true.

Key Concepts

Linear Model

A linear model is a mathematical representation of a relationship between two variables using a straight line. This is expressed in the equation format: \[y = ax + b\]where:

• \(y\) is the dependent variable
• \(x\) is the independent variable
• \(a\) is the slope of the line
• \(b\) is the y-intercept, the value of \(y\) when \(x\) is 0

The linear model is used to predict or explain the relationship between two variables. This type of model is simple yet powerful, especially when the relationship between the variables is approximately linear. Understanding the components of a linear model is crucial. It helps us interpret data, predict outcomes, and identify trends.

Slope Interpretation

The slope of a linear equation is a crucial component in understanding the relationship between two variables. It is represented by \(a\) in the equation \(y = ax + b\). The slope tells us how much \(y\) changes when \(x\) changes by one unit. A positive slope means that for every increase in \(x\), \(y\) also increases. This indicates a direct relationship. Conversely, a negative slope implies an inverse relationship, meaning as \(x\) increases, \(y\) decreases.In the model \(y = -0.238x + 25\), the slope \(-0.238\) specifies a decrease in \(y\) as \(x\) increases, thus pointing to a negative relationship. Therefore, by examining the slope, we can determine the nature of the relationship, whether it is direct, inverse, or may not exist at all.

Correlation Analysis

Correlation analysis is a method used to study the relationship between two variables. It helps us understand the strength and direction of that relationship. Correlation can be visualized using a linear model or statistically computed through a correlation coefficient, typically ranging from -1 to 1:

• A correlation of 1 indicates a perfect positive relationship.
• A correlation of 0 indicates no linear relationship.
• A correlation of -1 indicates a perfect negative relationship.

When analyzing the correlation in a linear model like \(y = -0.238x + 25\), the negative slope directly implies a negative correlation. This means as one variable increases, the other decreases. Understanding correlation is essential for making informed decisions, detecting patterns, and predicting future trends based on historical data.