2. Please hand in homework on Law of Large Numbers Dan Gilbert “Stumbling on Happiness”

3. Lecturer’s desk Physics- atmospheric Sciences (PAS) - Room 201 s c r e e n Row A Row B Row C Row D Row E Row F Row G Row H 131211109 87 Row A 14131211109 87 Row B 1514131211109 87 Row C 1514131211109 87 Row D 16 1514131211109 87 Row E 17 16 1514131211109 87 Row F 1716 1514131211109 87 Row G 1716 1514131211109 87 Row H 16 18 table Row A Row B Row C Row D Row E Row F Row G Row H 15141716 1819 16 15 18171920 17161918 2021 18172019 2122 19182120 2223 20192221 2324 18172019 2122 19182120 2223 2143 56 2143 56 2143 56 2143 56 2143 56 2143 56 2143 56 2143 56 Row J Row K Row L Row M Row N Row P 2143 5 2143 5 2143 5 2143 5 2143 5 1 5 Row J Row K Row L Row M Row N Row P 27262928 30 25242726 28 24232625 27 23222524 26 25242726 28 27262928 30 6 14 131211109 87 16151817 19 202122 614131211109 87 16 15 18 17 19 20212223 614131211109 87 16 15 18171920 2122 23 6 14 131211109 87 1624181719 20 2122 231525 6 14 131211109 87 1624181719 20 2122 231525 Row Q 2143 5 27262928 30 6 14 131211109 87 242223 21 - 15 25 37363938 40 34 3132 3335 69 87 13 table 14 18 192021

4. MGMT 276: Statistical Inference in Management Fall 2015

5. Please re-register your clicker http://student.turningtechnologies.com/

6. Just for Fun Assignments Go to D2L - Click on “Content” Click on “Interactive Online Just-for-fun Assignments” Please note: These are not worth any class points and are different from the required homeworks

7. Before our next exam (October 20 th ) OpenStax Chapters 1 – 11 Plous (10, 11, 12 & 14) Chapter 10: The Representativeness Heuristic Chapter 11: The Availability Heuristic Chapter 12: Probability and Risk Chapter 14: The Perception of Randomness Schedule of readings

8. On class website: Please print and complete homework worksheet #11 Due Thursday October 15 th Hypothesis Testing and Confidence Intervals Homework

9. By the end of lecture today 10/13/15 Confidence Intervals Logic of hypothesis testing Steps for hypothesis testing Levels of significance (Levels of alpha) what does p < 0.05 mean? what does p < 0.01 mean? One-tail versus Two-tail test Type I versus Type II Errors

11. Raw Scores Area & Probability Z Scores Formula z table Have raw score Find z Have z Find raw score Have area Find z Have z Find area Normal distribution Raw scores z-scores probabilities Building towards confidenc e intervals Par t 1

12. . Writing Assignment - Try this one: Please find the (2) raw scores that border exactly the middle 95% of the curve Mean of 100 and standard deviation of 5 Go to table.4750 nearest z = 1.96 mean + z σ = 100 + (1.96)(5) = 109.80 Go to table.4750 nearest z = -1.96 mean + z σ = 100 + (-1.96)(5) = 90.20 Building towards confidenc e intervals Par t 1

13. . Try this one: Please find the (2) raw scores that border exactly the middle 95% of the curve Mean of 30 and standard deviation of 2 Go to table.4750 nearest z = 1.96 mean + z σ = 30 + (1.96)(2) = 33.92 Go to table.4750 nearest z = -1.96 mean + z σ = 30 + (-1.96)(2) = 26.08 Building towards confidenc e intervals Par t 1

14. . Try this one: Please find the (2) raw scores that border exactly the middle 99% of the curve Mean of 30 and standard deviation of 2 Go to table.4950 nearest z = 2.58 mean + z σ = 30 + (2.58)(2) = 35.16 Go to table.4950 nearest z = -2.58 mean + z σ = 30 + (-2.58)(2) = 24.84 Building towards confidenc e intervals Par t 1

15. . Please find the raw scores that border the middle 99% of the curve Please find the raw scores that border the middle 95% of the curve Which is wider? Building towards confidenc e intervals Par t 1

16. . Please find the raw scores that border the middle 99% of the curve Please find the raw scores that border the middle 95% of the curve 95% Confidence Interval: We can be 95% confident that the estimated score really does fall between these two scores 99% Confidence Interval: We can be 99% confident that the estimated score really does fall between these two scores Building towards confidenc e intervals Par t 1

17. . Building towards confidenc e intervals Par t 2

18. Confidence Intervals (based on z): We are using this to estimate a value such as a population mean, with a known degree of certainty with a range of values The interval refers to possible values of the population mean. We can be reasonably confident that the population mean falls in this range (90%, 95%, or 99% confident) In the long run, series of intervals, like the one we figured out will describe the population mean about 95% of the time. Can actually generate CI for any confidence level you want – these are most common Standard Error of the Mean (SEM) Subjective vs Empirical Building towards confidenc e intervals Par t 3

19. Confidence Intervals (based on z): A range of values that, with a known degree of certainty, includes an unknown population characteristic, such as a population mean How can we make our confidence interval smaller? Decrease variability by increasing sample size Decrease level of confidence Decrease variability through more careful assessment and measurement practices (minimize noise). 95% Building towards confidenc e intervals Par t 3

20. . We now know all components of actually calculating confidence intervals: When to use confidence intervals: when you are estimating (guessing) a single number by providing likely range that the number appears in How to calculate confidence intervals Simply finding the raw score that is a certain distance from the mean that is associated with an area under the curve The relevance of the Central Limit Theorem When we are predicting a value we will use the standard error of the mean (rather than the standard deviation) Standard Error of the Mean (SEM)

21. 1) Go to z table - find z score for for area.4750 z = 1.96 Mean = 50 Standard deviation = 10 Find the scores for the middle 95% ? ?.9500.4750 ? 95% 2) x = mean + (z)(standard deviation) x = 50 + (-1.96)(10) x = 30.4 3) x = mean + (z)(standard deviation) x = 50 + (1.96)(10) x = 69.6 30.4 69.6 Please note: We will be using this same logic for “confidence intervals” x = mean ± (z)(standard deviation) Scores 30.4 - 69.6 capture the middle 95% of the curve

22. Mean = 50 Standard deviation = 10 n = 100 s.e.m. = 1 Find the scores for the middle 95% ? ?.9500.4750 ? 95% x = mean ± (z)(s.e.m.) x = 50 + (1.96)(1) x = 51.96 48.04 51.96 95% Confidence Interval is captured by the scores 48.04 – 51.96 Confidence intervals For “confidence intervals” same logic – same z-score But - we’ll replace standard deviation with the standard error of the mean standard error of the mean σ n = 10 100 = x = 50 + (-1.96)(1) x = 48.04

23. mean = 121 standard deviation= 15 n = 25 15 standard error of the mean σ n = raw score = mean + (z score)(standard error) x = x ± ( z )( σ x ) raw score = mean ± (z score)(sem) 100 110 120 130 140 25 =3 = X = 121 ± (1.96)(3) = 121 ± 5.88 (115.12, 126.88) confidence interval Please notice: We know the standard deviation and can calculate the standard error of the mean from it Find a 95% Confidence Interval for this distribution

24. Confidence intervals Tell me the scores that border exactly the middle 95% of the curve Construct a 95 percent confidence interval around the mean ?? 95% We know this Similar, but uses standard error the mean based on population s.d. raw score = mean ± (z score)(s.d.) raw score = mean ± (z score)(s.e.m.)

25. Confidence intervals Tell me the scores that border exactly the middle 95% of the curve - use z score of 1.96 Construct a 95 percent confidence interval around the mean z scores for different levels of confidence Level of Alpha 1.96 =.05 2.58 =.01 1.64 =.10 90% How do we know which z score to use?

26. Confidence Interval of 99% Has and alpha of 1% α =.01 Confidence Interval of 90% Has and alpha of 10% α =. 10 Confidence Interval of 95% Has and alpha of 5% α =.05 99%95%90% Area outside confidence interval is alpha Area in the tails is called alpha Area associated with most extreme scores is called alpha Critical z -2.58 Critical z 2.58 Critical z -1.96 Critical z 1.96 Critical z -1.64 Critical z 1.64

27. Moving from descriptive stats into inferential stats…. Measurements that occur outside this middle ranges are suspicious, may be an error or belong elsewhere Measurements that occur within the middle part of the curve are ordinary (typical) and probably belong there 99%95%90% Area outside confidence interval is alpha

28. How do we know if something is going on? How rare/weird is rare/weird enough? Every day examples about when is weird, weird enough to think something is going on? Handing in blue versus white test forms Psychic friend – guesses 999 out of 1000 coin tosses right Cancer clusters – how many cases before investigation Weight gain treatment – one group gained an average of 1 pound more than other group…what if 10?

29. Why do we care about the z scores that define the middle 95% of the curve? Inferential Statistics Hypothesis testing with z scores allows us to make inferences about whether the sample mean is consistent with the known population mean. Is the mean of my observed sample consistent with the known population mean or did it come from some other distribution?

30. Why do we care about the z scores that define the middle 95% of the curve? If the z score falls outside the middle 95% of the curve, it must be from some other distribution If a score falls out into the 5% range we conclude that it “must be” actually a common score but from some other distribution Main assumption: We assume that weird, or unusual or rare things don’t happen That’s why we care about the z scores that define the middle 95% of the curve

31. . I’m not an outlier I just haven’t found my distribution yet Main assumption: We assume that weird, or unusual or rare things don’t happen If a score falls out into the tails (low probability) we conclude that it “must be” a common score from some other distribution

32. ... 95% X Relative to this distribution I am unusual maybe even an outlier Relative to this distribution I am utterly typical Reject the null hypothesis Support for alternative hypothesis X

33. . Rejecting the null hypothesis If the observed z falls beyond the critical z in the distribution (curve): then it is so rare, we conclude it must be from some other distribution then we reject the null hypothesis then we have support for our alternative hypothesis If the observed z falls within the critical z in the distribution (curve): then we know it is a common score and is likely to be part of this distribution, we conclude it must be from this distribution then we do not reject the null hypothesis then we do not have support for our alternative not null x null big z score Alternative Hypothesis. x null small z score x x

34. Rejecting the null hypothesis If the observed z falls beyond the critical z in the distribution (curve): then it is so rare, we conclude it must be from some other distribution then we reject the null hypothesis then we have support for our alternative hypothesis If the observed z falls within the critical z in the distribution (curve): then we know it is a common score and is likely to be part of this distribution, we conclude it must be from this distribution then we do not reject the null hypothesis then we do not have support for our alternative hypothesis

35. How do we know how rare is rare enough? Critical z: A z score that separates common from rare outcomes and hence dictates whether the null hypothesis should be retained (same logic will hold for “critical t”) The degree of rarity required for an observed outcome to be “weird enough” to reject the null hypothesis Which alpha level would be associated with most “weird” or rare scores? Level of significance is called alpha ( α ) If the observed z falls beyond the critical z in the distribution (curve) then it is so rare, we conclude it must be from some other distribution 99%95%90% α =.01 α =.05 α =.10 Area in the tails is alpha

36. Rejecting the null hypothesis The result is “statistically significant” if: the observed statistic is larger than the critical statistic (which can be a ‘z” or “t” or “r” or “F” or x 2 ) observed stat > critical stat If we want to reject the null, we want our t (or z or r or F or x 2 ) to be big!! the p value is less than 0.05 (which is our alpha) p < 0.05 If we want to reject the null, we want our “p” to be small!! we reject the null hypothesis then we have support for our alternative hypothesis

37. Confidence Interval of 99% Has and alpha of 1% α =.01 Confidence Interval of 90% Has and alpha of 10% α =. 10 Confidence Interval of 95% Has and alpha of 5% α =.05 99%95%90% Critical z -2.58 Area in the tails is called alpha Critical z 2.58 Critical z -1.96 Critical z 1.96 Critical z -1.64 Critical z 1.64 Critical z separates rare from common scores

38. How would the critical z change? α = 0.05 Significance level =.05 α = 0.01 Significance level =.01 -1.96 or +1.96 -2.58 or +2.58 What if our observed z = 2.0? Reject the null Do not Reject the null Remember, reject the null if the observed z is bigger than the critical z Deciding whether or not to reject the null hypothesis.05 versus.01 alpha levels p < 0.05 Yes, Significant difference Not a Significant difference

39. How would the critical z change? α = 0.05 Significance level =.05 α = 0.01 Significance level =.01 -1.96 or +1.96 -2.58 or +2.58 What if our observed z = 1.5? Do Not Reject the null Do Not Reject the null Remember, reject the null if the observed z is bigger than the critical z Deciding whether or not to reject the null hypothesis.05 versus.01 alpha levels Not a Significant difference

40. How would the critical z change? α = 0.05 Significance level =.05 α = 0.01 Significance level =.01 -1.96 or +1.96 -2.58 or +2.58 What if our observed z = -3.9? Reject the null Remember, reject the null if the observed z is bigger than the critical z Deciding whether or not to reject the null hypothesis.05 versus.01 alpha levels p < 0.05 Yes, Significant difference p < 0.01 Yes, Significant difference

41. How would the critical z change? α = 0.05 Significance level =.05 α = 0.01 Significance level =.01 -1.96 or +1.96 -2.58 or +2.58 What if our observed z = -2.52? Reject the null Do not Reject the null Remember, reject the null if the observed z is bigger than the critical z Deciding whether or not to reject the null hypothesis.05 versus.01 alpha levels p < 0.05 Yes, Significant difference Not a Significant difference