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Please see and read the attachment.  Recall the following critical thinking skills: Inference, Deduction, Induction, Analysis, and Evaluation. In mathematics, each of these skills are important tools for coming to sound, reasonable answers to problems. Here we will focus on evaluation and analysis. Mathematical Modeling and Linear Regression: Interpolating, Extrapolating, and Analyzing Data Georgia Military College MAT109 – College Algebra QEP – Bright IDEA Student Learning Objectives 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! Name: ________________________________ MAT109 – College Algebra Mathematical Modeling: Interpolating, Extrapolating, and Analyzing Data. 0. Critical Thinking Review: Recall the following critical thinking skills: Inference, Deduction, Induction, Analysis, and Evaluation. In mathematics, each of these skills are important tools for coming to sound, reasonable answers to problems. Here we will focus on evaluation and analysis. __________________________________________________________________________________________Evaluation - Evaluative reasoning skills enable individuals to assess the credibility of sources of information and the claims they make. And, 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. __________________________________________________________________________________________ Analysis - identify assumptions, reasons and claims, and examine how they interact in the formation of arguments. Individuals use analytics to gather information from charts, graphs, diagrams, spoken language and documents. People with strong analytical skills attend to patterns and to details. They identify the elements of a situation and determine how those parts interact. Strong interpretations skills can support high quality analysis by providing insights into the significance of what a person is saying or what something means. __________________________________________________________________________________________ How do you think the critical thinking skills of evaluation and analysis apply to the content we have discussed so far into the term? When you work through a mathematical exercise, do you use any kind of analytical skills? How do you evaluate your solution? Can you think of a time when you were using a mathematical formula to arrive to an answer? How do you know this formula is correct? Write down some thoughts you have about evaluation and analysis in mathematics. Utilize anything you have learned from discussing with your classmates in the discussion forum. 1. Problem Introduction 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 below represents the measurements of different cultures (in thousands of bacteria) after 8 hours pH Number of Bacteria (thousands) 4 116 5 120 6 131 7 136 8 141 9 151 10 148 11 163 Create a scatter plot to represent the data: Based on the data and your scatter plot, what do you think a good linear function would be to represent the data? Justify your decision as best as you can. 2. Linear Regression & Problem Exploration The least-squares regression line is of the form , where and are defined as follows: Here, represents the number of data points given. With the data provided, find the least-squares regression line. What is the formula for the line you found? How does it compare to your original guess for a linear function to represent the data? 3. More Regression: The medical researcher decides to take a few more measurements to have a more complete data set. The following table represents the finalized data set detailing bacteria growth after 8 hours: pH Number of Bacteria (thousands) 3 124 4 116 5 120 6 131 7 136 8 141 9 151 10 148 11 163 12 155 With this data, calculate a new least-squares regression line. (your earlier work should help some) What is the formula you found? How does this compare to your original least-squares regression line? 4. Interpolation and Extrapolation Write down the formula you calculated in part 3: Using this formula, identify the amount of bacteria cultures that would grow in a pH level of 7.5. Include your calculations: Using the formula, identify the amount of bacteria cultures that would grow in a pH level of 1. Include your calculations: Now, using your formula from part 2, find these same values. How do they compare? Which do you believe is more accurate? Why? 5. Debriefing & Conclusions Write a brief response to some of the following thinking points: · Do you think there is a positive, negative, or no correlation between the pH level and the growth of bacteria? What calculations support your response? · From your initial guess to your two separate calculations of a least-squares regression line, how did your model change? Did your model better approximate some values versus others? Why do you think that is? · Suppose that the researcher made a mistake in calculating the data, and that at the pH value of 11, there were actually 263 thousand bacteria cultures. What affect do you think this would have on the least-squares regression line? · It isn’t appropriate to model all data with linear functions. Sometimes, quadratic functions or even other polynomial functions are used. Do you think that a linear function was an appropriate model here? Why or why not? Write a brief response addressing each of these thinking points: · How did you use the critical thinking skill of evaluation in this scenario? Did you use any evidence or justification to support any thoughts or calculations? · How did you use the critical thinking skill of analysis in this scenario? Did you analyze any charts, or recognize any relationships amongst data? · When interpolating, we are using deductive reasoning. When extrapolating, we are using inductive reasoning. What could the researcher do to evaluate your interpolations and extrapolations?