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Stat 31, Section 1, Last Time

This text introduces the foundations of statistical inference, including population vs. samples, numerical summaries, probability, and assessing uncertainty. It also discusses midterm exam results and correlation analysis.

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Stat 31, Section 1, Last Time

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  1. Stat 31, Section 1, Last Time • Foundations of Statistical Inference • Population vs. Samples • Numerical Summaries (mean, SD): • Population: “parameters”, • Sample: “statistics” • Probability: • Numerical Assessment of “uncertainty” • Makes statistics a Quantitative Science

  2. Midterm I Results No grade range given now, since “not enough info…” Recall earlier analyses: Final vs. Midterm I Midterm II vs. Midterm I Final Exam vs. Midterm I Correlation = 0.61 Correlation = 0.57 Correlation = 0.73

  3. Midterm I Results Indicator of Your Status: 90-100 “Very Pleased” 78-89 “OK, but be careful…” 0-77 “Strongly recommend drop course” (or let’s talk…)

  4. Midterm I Results

  5. Pepsi - Coke Results Summary Spreadsheet: https://www.unc.edu/~marron/UNCstat31-2005/Stat31CokePepsiResults2005.xls Prefer Pepsi? 51% Pepsi Sweeter? 53% Think Know? 82% Right? 69% How many Heads? 58% What is “random variation”????

  6. Probability Recall Basics: Assign numbers (representing “how likely”), to outcomes E.g. Die Rolling: P{comes up 4} = 1/6 • Outcome is “4” • Probability is 1/6

  7. Probability - Events More Terminology (to carry this further): • An event is a set of outcomes Die Rolling: “an even #”, is the event {2, 4, 6} Notes: • If betting on even don’t care about #, only even or odd • Thus events are our foundation • Each outcome is an event: the set containing just that outcome • So event is the more general concept

  8. Probability on Events Sample Space is the set of all outcomes = = “event with everything that can happen” Extend Probability to Events by: P{event} = sum of probs of outcomes in event

  9. Probability Technical Summary: • A probability model is a sample space • I.e. set of outcomes, plus a probability, P • P assigns numbers to events, • Events are sets of outcomes

  10. Probability Function The probability, P, is a “function”, defined on a set of events Recall function in math: plug-in get out Probability: P{event} = “how likely”

  11. Probability Function E.g. Die Rolling • Sample Space = {1, 2, 3, 4, 5, 6} • “an even #” is the event {2, 4, 6} (a “set”) • P{“even”} = P{2, 4, 6} = = P{2} + P{4} + P{6} = = 1/6 + 1/6 + 1/6 = 3/6 = ½ • Fits, since expect “even half the time”

  12. Probability HW HW: 4.11 4.13b 4.17 4.19a, b

  13. Probability Now stretch ideas with more interesting e.g. E.g. Political Polls, Simple Random Sampling 2 views: • Each individual equally likely to be in sample • Each possible sample is equally likely Allows for simple Probability Modelling

  14. Simple Random Sampling • Sample Space is set of all possible samples • An Event is a set of some samples E.g. For population A, B, C, D • Each is a voter • Only 4, so easy to work out

  15. S. R. S. Example For population A, B, C, D, Draw a S. R. S. of size 2 Sample Space = {(A,B), (A,C), (A,D), (B,C), (B,D), (C,D)} outcomes, i.e. possible samples of size 2

  16. S. R. S. Example Now assign P, using “equally likely” rule: P{A,B} = P{A,C} = … = P{C,D} = = 1/(#samples) = 1/6 An interesting event is: “C in sample” = {(A,C),(B,C),(D,C)} (set of samples with C in them)

  17. S. R. S. Example P{C in sample} = i.e. happens “half the time”. HW: 4.29

  18. Political Polls Example What is your chance of being in a poll of 1000, from S.R.S. out of 200,000,000? (crude estimate of # of U. S. voters) Recall each sample is equally likely so: Problem: this is really big (5,733 digits, too big for easy handling….)

  19. Political Polls Example More careful calculation: Makes sense, since you are “equally likely to be in samples”

  20. And now for something completely different .

  21. Probability • Now have prob. models • But still hard to work with • E.g. prob’s we care about, such as “accuracy estimators”, need better tools • Need to look more deeply

  22. 3 Big Rules of Probability • Main idea: calculate “complicated prob’s” • By decomposing events in terms of simple events • Then calculating probs of these • And then using simple rules of prob.

  23. 3 Big Rules of Probability Rule I: the not rule: P{not A} = 1 – P{A} Why? E.g. equally likely sample points: And more generally:

  24. The “Not” Rule of Probability Text Book Terminology (sec. 4.2): not A = for “complement” (set theoretic term) (I prefer “not”, since more intuitive)

  25. The “Not” Rule of Probability HW: Rework, using the “not” rule: 4.17b 4.19a,b

  26. 3 Big Rules of Probability Rule II: the or rule: P{A or B} = P{A} + P{B} – P{A or B} Why? E.g. equally likely sample points: Helpful Pic:

  27. Big Rules of Probability E.g. Roll a die, Let A = “4 or less” = {1, 2, 3, 4} Let B = “Odd” = {1, 3, 5} Check how rules work by calculating 2 ways: Direct: P{not A} = P{5, 6} = 2/6 = 1/3 By Rule I: P{not A} = 1 – P{A} = 1 – 4/6 = 1/3

  28. The “Or” Rule of Probability A = “4 or less” = {1, 2, 3, 4} B = “Odd” = {1, 3, 5} Check how rule works by calculating 2 ways: Direct: P{A or B} = P{1, 2, 3, 4, 5} = 5/6 By Rule II: P{A or B} = = P{A} + P{B} – P{A or B} = = 4/6 + 3/6 – 2/6 = 5/6 (check!)

  29. The “Or” Rule of Probability • Seems too easy? • Don’t really need rules for these simple things • But they are the key to bigger problems • Such as Simple Random Sampling HW:

  30. The “Or” Rule of Probability • Seems too easy? • Don’t really need rules for these simple things • But they are the key to bigger problems • Such as Simple Random Sampling HW: 4.86 (0.308)

  31. The “Or” Rule of Probability E.g: A college has 60% Women and 40% smokers, and 50% women who don’t smoke. What is the chance that a randomly selected student is either a women or a non-smoker? (seems “>60%”, but twice? Must be < 100%, i.e. must be some overlap…)

  32. College Women – Smokers E.g. P{W or S} = P{W} + P{S} = P{W & S} (choice of letters make easy to work with) = 0.6 + (1 – 0.4) – 0.5 = 0.7 (answer is 70% women or smokers) Note: rules are powerful when used together HW: 4.89

  33. The “Or” Rule of Probability E.g. Events A & B are “mutually exclusive”, i.e. “disjoint”, when P{A & B} = 0 (i.e. no chance of seeing both at same time) Useful Pic: Then: P{A or B} = P{A} + P{B} Text suggest “new rule”, I say “special case”

  34. The “Exclusive Or” Rule HW: 4.18 (0.65, 0.38, 0.62)

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