Stor 155, Section 2, Last Time

1. Stor 155, Section 2, Last Time • Prediction in Regression • Given new point X0, predict Y0 • Confidence interval for mean • Prediction Interval for value • Review…

2. Stat 31 Final Exam: Date & Time: Tuesday, May 8,  8:00-11:00 Last Office Hours: • Thursday, May 3, 12:00 - 5:00 • Monday, May 7, 10:00 - 5:00 • & by email appointment (earlier) Bring with you, to exam: • Single (8.5" x 11") sheet of formulas • Front & Back OK

3. Review Slippery Issues Major Confusion: Population Quantities Vs. Sample Quantities

4. Response to a Request You said at the end of today's class that you would be willing to take class time to "reteach" concepts that might still be unknown to us. Well, in my case, it seems that probability and probability distribution is a hard concept for me to grasp. On the first midterm, I missed … and on the second midterm, I missed … I seem to be able to grasp the other concepts involving binomial distribution, normal distribution, t-distribution, etc fairly well, but probability is really killing me on the exams. If you could reteach these or brush up on them I would greatly appreciate it.

5. Levels of Probability • Simple Events • Big Rules of Prob (Not, And, Or) • Bayes Rule • Distributions (in general) • Defined by Tables • Summary of discrete probs • Get probs by summing • Uniform • Get probs by finding areas

6. Levels of Probability • Distributions (in general) • Named (& Useful) Distributions • Binomial • Discrete distribution of Counts • Compute with BINOMDIST & Normal Approx. • Normal • Continuous distribution of Averages • Compute with NORMDIST & NORMINV • T • Similar to Normal, for estimated s.d. • Compute with TDIST & TINV

7. Detailed Look Simple Events: • Big Rules of Probability: • Not Rule ( 1 – P{opposite}) • Or Rule (glasses – football) • And rule (multiply conditional prob’s) • Use in combination for real power • Bayes Rule • Turn around conditional probabilities • Write hard ones in terms of easy ones • Recall surprising disease testing result

8. Detailed Look • Distributions (in general) • Defined by Tables • Summary of discrete probs • Get probs by summing • Easy to forget after so much other stuff… Studied in Notes: 2/20, 2/22, 3/1 Some highlights…

9. Highlights of Dist’ns in Tables • Distributions (in general) • Defined by Tables • Summary of discrete probs • Get probs by summing • Easy to forget after so much other stuff… Studied in Notes: 2/20, 2/22, 3/1 Some highlights…

10. Random Variables Die rolling example, for X = “net winnings”: Win $9 if 5 or 6, Pay $4, if 1, 2 or 4 Probability Structure of X is summarized by: P{X = 9} = 1/3 P{X = -4} = 1/2 P{X = 0} = 1/6 Convenient form: a table

11. Summary of Prob. Structure In general: for discrete X, summarize “distribution” (i.e. full prob. Structure) by a table: Where: • All are between 0 and 1 • (so get a prob. funct’n as above)

12. Summary of Prob. Structure Summarize distribution, for discrete X, by a table: Power of this idea: • Get probs by summing table values • Special case of disjoint OR rule

13. Summary of Prob. Structure E.g. Die Rolling game above: P{X = 9} = 1/3 P{X < 2} = P{X = 0} + P{X = -4} =1/6+1/2 = 2/3 P{X = 5} = 0 (not in table!)

14. Summary of Prob. Structure E.g. Die Rolling game above:

15. Mean of Discrete Distributions Frequentist approach to mean: a weighted average of values where weights are probabilities

16. Mean of Discrete Distributions E.g. Above Die Rolling Game: Mean of distribution = = (1/3)(9) + (1/6)(0) +(1/2)(-4) = 3 - 2 = 1 Interpretation: on average (over large number of plays) winnings per play = $1 Conclusion: should be very happy to play

17. Variance of Random Variables So define: Variance of a distribution As: random variable

18. Variance of Random Variables E. g. above game: =(1/2)*5^2+(1/6)*1^2+(1/3)*8^2 Note: one acceptable Excel form, e.g. for exam (but there are many)

19. Standard Deviation Recall standard deviation is square root of variance (same units as data) E. g. above game: Standard Deviation =sqrt((1/2)*5^2+(1/6)*1^2+(1/3)*8^2)

20. And Now for Something Completely Different Thought Provoking Movie… http://www.aclu.org/pizza/

21. Review Slippery Issues Major Confusion: Population Quantities Vs. Sample Quantities

22. Recall Pepsi Challenge In class taste test: • Removed bias with randomization • Double blind approach • Asked which was: • Better • Sweeter • which

23. Recall Pepsi Challenge Results summarized in http://stat-or.unc.edu/webspace/postscript/marron/Teaching/stor155-2007/Stat155CokePepsiResults2007.xls Recall Eyeball impressions: a. Perhaps no consensus preference between Pepsi and Coke? • Is 54% "significantly different from 50%? Result of "marketing research"???

24. Recall Pepsi Challenge b. Perhaps no consensus as to which is sweeter? • Very different from the past, when Pepsi was noticeably sweeter • This may have driven old Pepsi challenge phenomenon • Coke figured this out, and matched Pepsi in sweetness

25. Recall Pepsi Challenge c. Most people believe they know • Serious cola drinkers, because now flavor driven • In past, was sweetness driven, and there were many advertising caused misperceptions! d. People tend to get it right or not??? (less clear) • Overall 71% right. Seems like it, but again is that significantly different from 50%?

26. Recall Pepsi Challenge e. Those who think they know tend to be right??? • People who thought they knew: right 71% of the time f. Those who don't think they know seem to right as well. Wonder why? • People who didn't: also right 70% of time? Why? "Natural sampling variation"??? • Any difference between people who thought they knew, and those who did not think so?

27. Recall Pepsi Challenge g. Coin toss was fair (or is 57% heads significantly different from %50?) How accurate are those ideas? • Will build tools to assess this • Called “hypo tests” and “P-values” • Revisit this now

28. Pepsi – Coke Taste Test Data and Analysis: http://stat-or.unc.edu/webspace/postscript/marron/Teaching/stor155-2007/Stat155CokePepsiResults2007.xls Hypothesis Tests: • Proportions based (i.e. think about p) • Interesting Hypos: • Recall Sampling Distribution:

29. Pepsi – Coke Taste Test Data and Analysis: http://stat-or.unc.edu/webspace/postscript/marron/Teaching/stor155-2007/Stat155CokePepsiResults2007.xls P-value: P{what saw or m.c. | p = 0.5} Under assumption p = 0.5, So compute P-value as: Area obs’d

30. Pepsi – Coke Taste Test Data and Analysis: http://stat-or.unc.edu/webspace/postscript/marron/Teaching/stor155-2007/Stat155CokePepsiResults2007.xls Compute P-value as: Area obs’d =NORMDIST(ABS(phat – 0.5),0, 1/(2*SQRT(n),TRUE)

31. Pepsi – Coke Taste Test Conclusions (P-values): http://stat-or.unc.edu/webspace/postscript/marron/Teaching/stor155-2007/Stat155CokePepsiResults2007.xls • No consensus, Pepsi vs. Coke (0.46) • No consensus, Sweeter (0.81) • Most think know (e-5, very strong) • Get It Right (0.0006, very strong) • Fair Coin Toss (0.21, seems OK) • Thought Right, Were Right (0.003,yes) • Thought Not, Were Right (0.09, perhaps too modest?)

32. Pepsi – Coke Taste Test Some interesting history of this test: • First Attempts • Pepsi was preferred • Pepsi was sweeter • Many got it wrong (even if thought new) • Reason for “Pepsi challenge”? • New Coke Came Out • Response to Pepsi Challenge?

33. Pepsi – Coke Taste Test Some interesting history of this test: • New Coke Came Out • People thought they hated it… • Anger over changing the flavor… • So Coke Classic came out • Fun for me: New Coke vs. Coke Classic

34. Pepsi – Coke Taste Test Some interesting history of this test: • Taste test: New Coke vs. Coke Classic • New Coke preferred to Coke Classic! • New Coke was sweeter • Most got it wrong (even if thought new) • Changes Over Time • Appears Coke Classic slowly morphed into New Coke…