Introduction to Modeling Data with Statistics and Algebra

Have you ever wondered if there’s a connection between your homework scores and your exam performance? Or how about the relationship between study time and overall GPA? These questions are just the beginning of what we can explore with data analysis.

In this chapter, we’ll dive into the world of bivariate data – examining how two numeric variables relate to each other. While real-world problems often involve multiple variables (multivariate data), we’ll start with the fundamentals: just two variables.

We’ll focus on two types of relationships:

1. Linear relationships: Imagine hours worked and wages earned in a job with a fixed hourly rate.
2. Exponential relationships: Picture the growth curve of a viral social media post – starting slow, then exploding in popularity.

Using both algebra and regression analysis, we’ll learn how to describe these relationships and measure their strength (for linear). This skill set is invaluable across various fields:

1. Education: Analyzing the impact of class size on student performance
2. Public Health: Modeling the spread of infectious diseases like COVID-19
3. Economics: Examining the relationship between interest rates and inflation
4. Climate Science: Studying the correlation between carbon emissions and global temperatures

By the end of this chapter, you’ll have the tools to:

1. Recognize patterns in paired data sets
2. Apply statistical techniques to real-world problems
3. Use algebraic models to make predictions about related variables in real-world scenarios

These skills form the foundation for more advanced statistical analysis and are essential for evidence-based decision making in many professional fields. As you progress, you’ll see how these basic principles extend to more complex multivariate analyses used in advanced research and data science.

Learning Objectives

Below are the learning objectives for each section of the chapter.

5.1 Modeling Linear Relationships with Algebra

• Identify elements in a linear model of the form $y=mx+b$
• Create a linear model with algebra between two quantitative variables
• Graph a linear model
• Solve application problems using a linear model created with algebra

5.2 Modeling Exponential Relationships with Algebra

• Identify elements in an exponential model of the form $y=a(b^x)$
• Create an exponential model with algebra between two quantitative variables
• Graph an exponential model
• Solve application problems using an exponential model created with algebra

5.3 Modeling Linear Relationships with Regression

• Create a Scatterplot
• Calculate the correlation coefficient to describe the linear relationship between two quantitative variables
• Create a linear regression model between two quantitative variables using a spreadsheet
• Interpret a linear regression model in applications

5.4 Modeling Exponential Relationships with Regression

• Create an exponential regression model between two quantitative variables using a spreadsheet
• Interpret an exponential regression model in applications

Attributions

• This page contains modified content from “OpenStax Introductory Satistics” by Barbara Illowsky, Susan Dean. Licensed under CC BY 4.0.
• This page contains content by Robert Foth, Math Faculty, Pima Community College, 2024.