After reviewing the following information, you will download and complete the artifact document (which can be found by clicking on this

link

). You can either fill it out by hand and scan it (including saving and uploading pictures) or you can type it all in. If you elect to type all your information in, your mathematical notation must be correct.

Name: ________________________________

MAT109 β 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

4

5

6

7

8

9

10

11

Number of Bacteria (thousands)

116

120

131

136

141

151

148

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 π¦ = ππ₯ + π, where π and π are defined as follows:

π=

π(Ξ£π₯π¦) β (Ξ£π₯)(Ξ£π¦)

π(Ξ£π₯ 2 ) β (Ξ£π₯)2

(Ξ£π¦)(Ξ£π₯ 2 ) β (Ξ£π₯)(Ξ£π₯π¦)

π=

π(Ξ£π₯ 2 ) β (Ξ£π₯)2

Here, π 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

3

4

5

6

7

8

9

10

11

12

Number of Bacteria (thousands)

124

116

120

131

136

141

151

148

163

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:

β’

β’

β’

β’

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 isnβt 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:

β’

β’

β’

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?