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.

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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?