Reasoning with DataChapter 2 2019 McGraw-Hill Education- All rights re.docx
- 1. Reasoning with Data Chapter 2 © 2019 McGraw-Hill Education. All rights reserved. Authorized only for instructor use in the classroom. No reproduction or distribution without the prior written consent of McGraw-Hill Education. Learning Objectives Define reasoning. Execute deductive reasoning. Explain an empirically testable conclusion. Execute inductive reasoning. Differentiate between deductive and inductive reasoning. Explain how inductive reasoning can be used to evaluate an assumption. Describe selection bias in inductive reasoning. ‹#› ‹#› © 2019 McGraw-Hill Education. What is Reasoning? Reasoning is the process of forming conclusions, judgments, or inferences from facts or data Reasoning and logic are often used interchangeably Logic is a description of the rules and/or steps behind the reasoning process ‹#› ‹#› © 2019 McGraw-Hill Education. Two Arguments
- 2. Argument 1: The companies profits are up more than 10% over the past year. An increase in profits of 10% is the result of excellent management. You were the manager over the past year. Therefore, I conclude that you engaged in excellent management last year. Argument 2: Ten of your 300 employees came to me with complaints about your management. They indicated that you treated them unfairly by not giving them a raise they deserved. Therefore, I conclude that all of your employees are disgruntled with your management. ‹#› ‹#› © 2019 McGraw-Hill Education. Understanding Reasoning In presenting the two arguments, the goal is not to make a definitive decision about which you believe (if either) The goal is to think about and distinguish different “lines― of reasoning In distinguishing between the different types of reasoning, you will be able to establish why you believe or question the claims made in the two arguments ‹#› ‹#› © 2019 McGraw-Hill Education. Two Major Types of Reasoning Reasoning Deductive Reasoning Inductive Reasoning Both play an important role in interpreting and drawing conclusions from data analysis ‹#› ‹#›
- 3. © 2019 McGraw-Hill Education. Deductive Reasoning Deductive Reasoning Goes from the general to the specific Also known as top-down logic Seeks to prove statements of the form “If A, then B― ‹#› ‹#› © 2019 McGraw-Hill Education. Deductive Reasoning Such reasoning always implies three underlying components: assumptions (“If A―), methods of proof (“thenâ€), and conclusions (“Bâ€) ‹#› ‹#› © 2019 McGraw-Hill Education. Deductive Reasoning The purest applications of deductive reasoning are in the field of mathematics Two of the most used approaches are direct proofs and transposition Direct proofs Proof that begins with assumptions, explains methods of proof, and states the conclusion(s) Transposition Any time a group of assumptions implies a conclusion, then it is also true that any time the conclusion does not hold, at least one of the assumptions must not hold ‹#›
- 4. ‹#› © 2019 McGraw-Hill Education. Direct Proof Let’s prove the following statement by direct proof: If X and Y are odd numbers, then their sum (X + Y) is an even number An Example: If X = 5(odd) and Y = 9(odd), then their sum X + Y = 14 is an even number Failing to find a contradiction is not the same a proving a statement is generally true ‹#› ‹#› © 2019 McGraw-Hill Education. Direct Proof: A Mathematical Approach If X and Y are odd numbers, then their sum (X + Y) is an even number X and Y are odd numbers If X is an odd number, then X can be written as X=2K+1, where K is an integer. (Example: X=13 X=(2 × 6)+1) If Y is an odd number, then Y can be written as Y =2M+1, where M is an integer, (Example: Y=23 Y=(2 × 11)+1) X+Y=(2K+1)+(2M+1)=2K+2M+2=2(K+M+1) K+M+1 is an integer so X+Y is 2 times an integer Any number that is 2 times an integer is divisible by 2 This means X+Y is even ‹#› ‹#›
- 5. © 2019 McGraw-Hill Education. Direct Proof: Common Sense Approach “If McDonald’s offers breakfast all day, their revenues will increase.― McDonald’s stores offer breakfast all day. The addition of breakfast during lunch/dinner hours implies more choices. Customers already choosing McDonald’s during lunch/dinner hours can continue buying the same meals at McDonald’s. Customers not choosing McDonald’s during lunch/dinner hours may start eating at McDonald’s. Retaining current customers and adding new ones, McDonald’s revenues will increase overall. ‹#› ‹#› © 2019 McGraw-Hill Education. Transposition While direct proofs are sufficient to prove a point logically, an alternative approach, transposition, may be more effective Transposition Is the equivalence between the statements “If A, then B― and “If not B, then A― Any time a group of assumptions implies a conclusion, then it is also true that any time the conclusion does not hold, then at least one of the assumptions must not hold ‹#› ‹#› © 2019 McGraw-Hill Education. Transposition “If A, then B†AND “If not B, then not Aâ€
- 6. ASSUMPTIONS (A) CONCLUSIONS (B) ‹#› ‹#› © 2019 McGraw-Hill Education. Transposition: A Mathematical Approach Prove the statement: If X2 is even, then X is even Suppose X is not an even number; it is instead an odd number If X is an odd number, then X= (2K +1), where K is an integer X2 = (2K+1)2 = 4K2 + 4K+1. 4K2 + 4K = 4(K2 +K) and so is divisible by 2 4K2 + 4K is an even number X2 = 4(K2 +K)+1 is an even number plus 1, meaning it is an odd number ‹#› ‹#› © 2019 McGraw-Hill Education. Transposition The statement was: If X2 is even, then X is even Using transposition, the opposite of the conclusion is used to proof the opposite of the assumption: If X is odd, then X2 is even would be incorrect Transposition can also be used without using mathematics to prove statements like “If A, then B―.
- 7. Transposition can be particularly effective if an assumption seems indisputably obvious. ‹#› ‹#› © 2019 McGraw-Hill Education. Transposition: An Example Proof the statement: “If McDonald’s stores offer breakfast all day, revenue will increase― McDonald’s stores revenues will not increase This means total revenues from current and new customers will not increase This means either there will be no new customers or revenues from current customers will decrease This means there could not have been an expansion in the menu McDonald’s stores do not offer breakfast all day ‹#› ‹#› © 2019 McGraw-Hill Education. Direct Proof and Transposition Direct Proof State assumptions Explain methods of proof (mathematics, common sense, etc.) State conclusions Transposition Assume the opposite of the conclusion Explain methods of proof (mathematics, common sense, etc.)
- 8. State assumption(s) that is (are) violated (not A) ‹#› ‹#› © 2019 McGraw-Hill Education. Deductive Reasoning Used commonly in the application of law If there is disagreement with a conclusion there are two possible sources: The method of proof, OR The assumption There are two ways of resolving disputes about assumptions Show robustness- the persistent accuracy of a conclusion despite variation in the associated assumption(s) within the context of a deductive argument Assess consistency with a collected dataset ‹#› ‹#› © 2019 McGraw-Hill Education. Empirically Testable Conclusions An empirically testable conclusion is a conclusion whose validity can be meaningfully tested using observable data. Example: A banana company’s management staffs are divided into two groups about their product’s placement in a major grocery store chain. Group 1 believes that change in current location will increase its sales. Group 2 believes that current location is good enough. ‹#›
- 9. ‹#› © 2019 McGraw-Hill Education. Empirically Testable Conclusions Company has the sales data in the current location. Company chooses to move its product to a new location and collects sales data. Now the company can meaningfully test the validity of the management’s competing conclusions. Making the actual decision about the validity of an empirically testable conclusion based on observable data is an application of inductive reasoning ‹#› ‹#› © 2019 McGraw-Hill Education. Inductive Reasoning Inductive reasoning Reasoning that goes from the specific to the general; bottom-up logic Population The entire set of potential observations about which we want to learn Data sample A subset of population that is collected and observed ‹#› ‹#› © 2019 McGraw-Hill Education. Inductive Reasoning Business regularly collect data samples to draw conclusions about the population after applying inductive reasoning.
- 10. The conclusion from inductive reasoning requires degree of support (also called inductive probability). Degree of support is also called the strength of the inductive argument. Example: if we are 50% confident about the conclusion, then the degree of support is 50%. ‹#› ‹#› © 2019 McGraw-Hill Education. Degrees of Support Two Types of Degrees of Support Both play an important role in interpreting and drawing conclusions from data analysis SUBJECTIVE DEGREE OF SUPPORT (IT IS BASED ON OPINION AND LACKING STATISTICAL FOUNDATION) OBJECTIVE DEGREE OF SUPPORT (IT HAS A STATISTICAL FOUNDATION AND THUS MORE CREDIBLE THAN SUBJECTIVE DEGREE OF SUPPORT) ‹#› ‹#› © 2019 McGraw-Hill Education. Evaluating Assumptions Through deductive reasoning, an empirically testable conclusion is made Collect a data sample Test the conclusion by comparing the observed outcomes in the data samples to their corresponding probabilities Use inductive reasoning to decide whether the conclusion passes or fails If it fails, transposition implies we must reject
- 11. If it passes, then we must not reject ‹#› ‹#› © 2019 McGraw-Hill Education. Inductive Reasoning for Evaluating Assumptions ‹#› ‹#› © 2019 McGraw-Hill Education. Selection Bias in Inductive Reasoning Improper use of inductive reasoning may lead to inaccurate, or biased conclusions Data-generating process is typically the source of the bias Survey questions constructed in a leading way Confirmation bias is the tendency to confirm a claim Predictable patterns are discovered Predictable-world bias is the tendency to find order when none exists, and occurs when people “read too much― into perceived patterns from random data ‹#› ‹#› © 2019 McGraw-Hill Education. Selection Bias Selection bias The act of drawing conclusions about a population using a selected data sample, without accounting for the means of selection There are two common types:
- 12. Collector selection bias occurs when the collector selects the members of the data sample in a systematic way Availability bias occurs when the collector of the data sample selects the members of the data sample according to what is most readily available Member selection bias occurs when potential members of the data sample self-select into, or out of, the sample ‹#› ‹#› © 2019 McGraw-Hill Education. image1.png image2.png image3.JPG