PS400 Quantitative Methods

2. PS400 Quantitative Methods • Dr. Robert D. Duval • Course Introduction • Presentation Notes and Slides • Version of January 9, 2001

3. Overview of Course P Syllabus P Texts P Grading P Assignments P Software

4. The First Two Weeks • P Review and Setting • The Logic of Research • P Logic • P Microcomputers • P Statistics

5. Overview of Statistics P Descriptive Statistics P Frequency Distributions P Probability P Statistical Inference P Statistical tests P Contingency Tables P Regression Analysis

6. The Logic of Research A quick review of the research process P Theory P Hypothesis P Observation P Analysis

7. Sample Theories • IR - Balance of Power • Wars erupt when there are shifts in the balance of power • Domestic Policy • The crime rate is affected by the economy

9. Theory

10. Theory Hypothesis

11. Theory Hypothesis Observation

12. Theory Analysis Hypothesis Observation

13. Theory Analysis Hypothesis Observation

14. Theory Deduction Analysis Hypothesis InductionOperationalization Observation Confirmation/ rejection

16. Logic A short primer on Deduction and Inference We will look at Symbolic Logic in order to examine how we employ deduction in cognition.

17. Logic What is Logic? • Logic • The study by which arguments are classified into good ones and bad ones. • Comprised of Statements • "Roses are red“ • "Republicans are Conservatives“

18. Logic Compound Statements • Conjunctions (Conjunction Junction) • Two simple statements may be connected with a conjunction • “and” • "Roses are Red and Violets are blue.“ • "Republicans are conservative and Democrats are liberal.“ • “or” • "Republicans are conservative or Republicans are moderate."

19. Operators • There are three main operators • And (•) • Or (v) • Not (~) • These may be used to symbolize complex statements • The other symbol of value is • Equivalence () • This is not quite the same as “equal to”.

20. Truth Tables • Statements have “truth value” • For example, take the statement P•Q: • This statement is true only if P and Q are both true. P Q P•Q T T T T F F F T F F F F

21. Truth Tables (cont) • Hence “Republicans are conservative and Democrats are liberal.” is true only if both parts are true. • On the other hand, take the statement PvQ: • This statement is true only if either P or Q are true, but not both. (Called the “exclusive or”) P Q PvQ T T F T F T F T T F F F

22. The Inclusive ‘or’ • Note that ‘or’ can be interpreted differently. • Both parts of the conjunction may be true in the “inclusive or”. This statement is true if either or both P or Q are true. P Q PvQ T T T T F T F T T F F F

23. The Inclusive ‘or’ • Note that ‘or’ can be interpreted differently. • Both parts of the conjunction may be true in the “inclusive or”. This statement is true if either or both P or Q are true. P Q PvQ T T T T F T F T T F F F

24. Tautologies • Note that p v ~p must be true • “Roses are red or roses are not red.” must be true. • A statement which must be true is called a tautology. • A set of statements which, if taken together, must be true is also called a tautology (or tautologous). • Note that this is not a criticism.

25. The Conditional • The Conditional • if a (antecedent) • then b (consequent) • It is also called the hypothetical, or implication. • This translates to: • A implies B • If A then B • A causes B

26. The Implication • We symbolize the implication by • We use the conditional or implication a great deal. • It is the core statement of the scientific law, and hence the hypothesis.

27. Equivalency of the Implication • Note that the Implication is actually equivalent to a compound statement of the simpler operators. • ~p v q • Please note that the implication has a broader interpretation than common English would suggest

28. Rules of Inference • In order to use these logical components, we have constructed “rules of Inference” • These rules are essentially “how we think.”

29. Disjunctive Syllogism

30. Hypothetical Syllogism

31. Modus Ponens

32. Modus Tollens

33. Logical Systems • Logic gives us power in our reasoning when we build complex sets of interrelated statements. • When we can apply the rules of inference to these statements to derive new propositions, we have a more powerful theory.

34. Tautologous systems • Systems in which all propositions are by definition true, are tautologous. • Balance of Power • Why do wars occur? Because there is a change in the balance of power. • How do you know that power is out of balance? A war will occur. • Note that this is what we typically call circular reasoning. • The problem isn’t the circularity, it is the lack of utility.

35. Paradoxes • P The Liars Paradox • < Epimenedes the Cretan says that all Cretans are liars.“ • P The ??? Paradox (a variant) • < The next statement is true. • < The previous statement is false.

36. Microcomputer Architecture

37. Using the Computer

38. Computers - Basic Architecture

39. Basic components

40. The CPU

41. Some simple binary arithmetic

42. Binary numbering

43. Binary addition

44. Miscellaneous

45. Digital Systems • So, in the end, we can see that computers simply move ad add 0’s and 1’s. • And out of this, we can build incredibly rich and complex experiences • Such as**** • Or…

47. Statistics A Philosophical Overview • Methods as Theory • Methods as Language

48. Principle organizing concepts P The Nature of the Problem P Measurement P Standards for comparison

49. Mathematical notation Important mathematical notation the student needs to know. n å X • PSummation • < For instance, the sum of all Xi from {I=1} to n means: beginning with the first number in your data set, add together all n numbers. • < The 3 is a symbolic representation of the process of adding up a specified series or collection of numbers. i i = 1

50. Mathematical notation – (cont.) P Square Roots and Exponents P e - the base of natural logarithms P Exponential and Logarithmic Equations