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Mathematical Modeling and Linear Regression: Interpolating, Extrapolating, and Analyzing Data

Mathematical Modeling and Linear Regression: Interpolating, Extrapolating, and Analyzing Data

Georgia Military College MAT109 – College Algebra QEP – Bright IDEA

Student Learning Objectives

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Understand, evaluate, and classify elementary mathematical functions.

Communicate mathematical ideas with clarity and coherence through both speaking and writing, to include the description, interpretation, and application of functions to real-world phenomena.

Analyze multiple step problems by incorporating a variety of problem-solving strategies, including algebraically, numerically, graphically, and with the use of appropriate technology.

Demonstrate the ability to manipulate mathematical concepts using symbolic (algebraic), numeric, and graphical means.

Critical Thinking – Evaluation

Evaluation: the process of judging whether an argument’s conclusion is logical or not.

Look at evidence that is presented and decide if it supports the conclusion.

Use the right tool for the argument: if deductive logic is being used, evaluate if the premises are sound enough to lead to the conclusion. If the argument is inductive, evaluate if the conclusion is based on the premises.

Critical Thinking – Evaluation

Evaluation – Evaluative reasoning skills enable individuals to assess the credibility of sources of information and the claims they make. People can use these skills to determine the strengths and weaknesses of arguments. Evaluation skills can judge the quality of analysis, interpretations, explanations, inferences, options, opinions, beliefs, ideas, proposals, and decisions. Strong explanation skills can support high quality evaluation by providing evidence, reasons, methods, criteria, or assumptions behind the made claims and reached conclusions.

 

Critical Thinking – Analysis

Analysis: A detailed examination of the structural form and content of some subject or situation, particularly as it relates to reasoning and interpretation.

Strong analytical skills are helpful in identifying patterns, important details, critical elements of a situation, and the interaction of such elements.

Analytical skills are used to examine data, charts, graphs, documents, and diagrams.

Critical Thinking – Analysis

Analysis: this is the ability to identify assumptions, reasons and claims, and examine how they interact in the formation of arguments. Strong interpretation skills can support high quality analysis by providing insights into the significance of what a person is saying or what something means.

 

Think about how evaluation and analysis applies to mathematics. Write down some of your thoughts! This will help you later in discussing with the class, and completing the Critical Thinking Assignment Artifact.

Mathematical Modeling

A mathematical model describes some phenomenon or system of phenomena, utilizing mathematical concepts and language. These phenomena could exist in fields of chemistry, physics, finance, psychology, and so on.

We use mathematics to describe the world around us, and based on the mathematics we develop, draw conclusions about the world.

Scatter Plots

Scatter plots are a useful way to represent data. We can easily depict different types of correlation between two variables, but sometimes we want to be able to draw stronger conclusions.

Consider the following:

A medical researcher wants to determine the effect of pH (a measure of alkalinity or acidity, with pure water having a pH of 7) on the growth of a bacteria culture. The data provided to you represents the measurements of different cultures (in thousands of bacteria) after 8 hours.

 

 

 

Regression Analysis

To find the “best” approximation, we need to find the least-squares regression line. This line will minimize the sum of the squares of the vertical distances of all data points from the line. This results in a linear equation (function) that will generate the least overall error, which is particularly important when we use our function to make predictions.

How can we calculate least-squares regression line?

The least-squares regression line is given by

 

 

 

 

With the data provided, you will find the least-squares regression line.

 

What will you need to do?

Now that you have watched this video, you should next head to the discussion forum and speak with your classmates about some of your ideas.

You should then download the QEP Assignment Artifact. The easiest way to complete this will be to download and print the artifact, write in your answers, and then scan and submit the completed document.

Feel free to return to the discussion forum as you complete the assignment to discuss the problems with your peers. The goal here is to utilize critical thinking skills!

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