1. Stat 31, Section 1, Last Time Distribution of Sample Means –Expected Value  same –Variance  less, Law of Averages, I –Dist’n  Normal, Law of Averages, II Statistical Inference –Confidence Intervals

2. Choice of Sample Size Additional use of margin of error idea Background: distributions Small n Large n

3. Choice of Sample Size Could choose n to make = desired value But S. D. is not very interpretable, so make “margin of error”, m = desired value Then get: “ is within m units of, 95% of the time”

4. Choice of Sample Size Given m, how do we find n? Solve for n (the equation):

5. Choice of Sample Size Graphically, find m so that: Area = 0.95 Area = 0.975

6. Choice of Sample Size Thus solve:

7. Choice of Sample Size EXCEL Implementation: Class Example 20, Part 3: https://www.unc.edu/~marron/UNCstat31-2005/Stat31Eg20.xls HW: 6.19, 6.21

8. Interpretation of Conf. Intervals 2 Equivalent Views: Distribution Distribution 95% pic 1 pic 2

9. Interpretation of Conf. Intervals Mathematically: pic 1 pic 2 no pic

10. Interpretation of Conf. Intervals Frequentist View: If repeat the experiment many times, About 95% of the time, CI will contain (and 5% of the time it won’t)

11. Interpretation of Conf. Intervals A nice Applet, from Ogden and West: http://www.amstat.org/publications/jse/v6n3/applets/ConfidenceInterval.html Try a few at “more interval” allows regeneration “on average” about 2.5/50 don’t cover This is idea of “% coverage”

12. Interpretation of Conf. Intervals Revisit Class Example 16 https://www.unc.edu/~marron/UNCstat31-2005/Stat31Eg16.xls Recall Class HW: Estimate % of Male Students at UNC CI View: Class Example 21 https://www.unc.edu/~marron/UNCstat31-2005/Stat31Eg21.xls Illustrates idea: CI should cover 95% of time

13. Interpretation of Conf. Intervals Class Example 21: https://www.unc.edu/~marron/UNCstat31-2005/Stat31Eg21.xls Q1: SD too small  Too many cover Q2: SD too big  Too few cover Q3: Big Bias  Too few cover Q4: Good sampling  About right Q5: Simulated Bi  Shows “natural var’n”

14. Interpretation of Conf. Intervals HW: 6.23, 6.26 (0.857, 0.135, 0.993)

15. Sec. 6.2 Tests of Significance = Hypothesis Tests Big Picture View: Another way of handling random error I.e. a different view point Idea: Answer yes or no questions, under uncertainty (e.g. from sampling of measurement error)

16. Hypothesis Tests Some Examples: Will Candidate A win the election? Does smoking cause cancer? Is Brand X better than Brand Y? Is a drug effective? Is a proposed new business strategy effective? (marketing research focuses on this)

17. Hypothesis Tests E.g. A fast food chain currently brings in profits of $20,000 per store, per day. A new menu is proposed. Would it be more profitable? Test: Have 10 stores (randomly selected!) try the new menu, let = average of their daily profits.

18. Fast Food Business Example Simplest View: for : new menu looks better. Otherwise looks worse. Problem: New menu might be no better (or even worse), but could have by bad luck of sampling (only sample of size 10)

19. Fast Food Business Example Problem: How to handle & quantify gray area in these decisions. Note: Can never make a definite conclusion e.g. as in Mathematics, Statistics is more about real life… (E.g. even if or, that might be bad luck of sampling, although very unlikely)

20. Hypothesis Testing Note: Can never make a definite conclusion, Instead measure strength of evidence. Approach I: (note: different from text) Choose among 3 Hypotheses: H + : Strong evidence new menu is better H 0 : Evidence in inconclusive H - : Strong evidence new menu is worse

21. Hypothesis Testing Terminology: H 0 is called null hypothesis Setup: H +, H 0, H - are in terms of parameters, i.e. population quantities (recall population vs. sample)

22. Fast Food Business Example E.g. Let = true (over all stores) daily profit from new menu. H + : (new is better) H 0 : (about the same) H - : (new is worse)

23. Fast Food Business Example Base decision on best guess: Will quantify strength of the evidence using probability distribution of E.g.  Choose H +  Choose H 0  Choose H -

24. Fast Food Business Example How to draw line? (There are many ways, here is traditional approach) Insist that H + (or H - ) show strong evidence I.e. They get burden of proof (Note: one way of solving gray area problem)

25. Fast Food Business Example Assess strength of evidence by asking: “How strange is observed value, assuming H 0 is true?” In particular, use tails of H 0 distribution as measure of strength of evidence

26. Fast Food Business Example Use tails of H 0 distribution as measure of strength of evidence: distribution under H 0 observed value of Use this probability to measure strength of evidence

27. Hypthesis Testing Define the p-value, for either H + or H 0, as: P{what was seen, or more conclusive | H 0 } Note 1: small p-value  strong evidence against H 0, i.e. for H + (or H - ) Note 2: p-value is also called observed significance level.

28. Fast Food Business Example Suppose observe:, based on Note, but is this conclusive? or could this be due to natural sampling variation? (i.e. do we risk losing money from new menu?)

29. Fast Food Business Example Assess evidence for H + by: H + p-value = Area

30. Fast Food Business Example Computation in EXCEL: Class Example 22, Part 1: https://www.unc.edu/~marron/UNCstat31-2005/Stat31Eg22.xls P-value = 0.094