2. BNAD 276: Statistical Inference in Management Winter, 2015 http://www.youtube.com/watch?v=Ahg6qcgoay4&watch_response Green sheet Seating Chart

3. Daily group portfolios Beginning of each lecture (first 5 minutes) Meet in groups of 3 or 4 Meet in groups of 3 or 4 Quiz one another on class material Quiz one another on class material Discuss the questions and determine the correct answer for each question Discuss the questions and determine the correct answer for each question Five copies (one for each group member – and typed) 3 multiple choice questions based on lecture Five copies (one for each group member – and typed) 3 multiple choice questions based on lecture Include 4 options (a, b, c, and d) Include 4 options (a, b, c, and d) Include a name and describe a person in a certain situation Include a name and describe a person in a certain situation Margaret was interested in taking a Statistics course. It is likely she was interested in studying which of the following? a. economic theories of communism b. theological perspectives of life after death c. musical compositions of the 12th century d. statistical techniques and inference They can be funny or serious, and must be clear and have only one correct answer.

5. . Homework Assignment Go to D2L - Click on “Content” Click on “Interactive Online Homework Assignments”

6. 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

7. 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

8. 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?

9. 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?

10. 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

11. . 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

12. ... 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

13. . 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

14. 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

15. 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

16. 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

17. 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

18. 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

19. 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

20. 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

21. 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

22. Five steps to hypothesis testing Step 1: Identify the research problem (hypothesis) Describe the null and alternative hypotheses Step 2: Decision rule Alpha level? ( α =.05 or.01)? Step 3: Calculations of “observed z” score Step 4: Make decision whether or not to reject null hypothesis If observed z is bigger then critical z then we reject null Step 5: Conclusion - tie findings back in to research problem 3 parts: 1) IV and DV and means (descriptive stats) 2) Which type of test and whether it was a significant difference 3) Summary results phrase Critical z statistic?

23. Critical Values What percent of the distribution will fall in region of rejection 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 For scores that fall into the regions of rejection, we reject the null 90% For scores that fall into the middle range, we do not reject the null 5% Moving from descriptive stats into inferential stats…. http://www.youtube.com/watch?v=0r7NXEWpheg http://today.msnbc.msn.com/id/33411196/ns/today-today_health/ Critical z -1.64 Critical z 1.64

24. Rejecting the null hypothesis The result is “statistically significant” if: the observed statistic is larger than the critical statistic 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 A note on decision making following procedure versus being right relative to the “TRUTH”

25. . Decision making: Procedures versus outcome Best guess versus “truth” What does it mean to be correct? Why do we say: “innocent until proven guilty” “not guilty” rather than “innocent” Is it possible we got a verdict wrong?

26. .. We make decisions at Security Check Points

27. .. Type I or Type II error? Does this airline passenger have a snow globe? Null Hypothesis means she does not have a snow globe (that nothing unusual is happening) – Should we reject it???!! As detectives, do we accuse her of brandishing a snow globe?

28. . Decision made by experimenter Status of Null Hypothesis (actually, via magic truth-line) Reject H o “yes snow globe, stop!” Do not reject H o “no snow globe move on” True H o No snow globe False H o Yes snow globe You are right! Correct decision You are wrong! Type I error (false alarm) You are right! Correct decision You are wrong! Type II error (miss) Are we correct or have we made a Type I or Type II error? Does this airline passenger have a snow globe? Note: Null Hypothesis means she does not have a snow globe (that nothing unusual is happening) – Should we reject it???!!

29. . Type I or type II error? Does this airline passenger have a snow globe? Type I error: Rejecting a true null hypothesis Saying the she does have snow globe when in fact she does not (false alarm) Type II error: Not rejecting a false null hypothesis Saying she does not have snow globe when in fact she does (miss) What would null hypothesis be? This passenger does not have any snow globe Two ways to be correct: Say she does have snow globe when she does have snow globe Say she doesn’t have any when she doesn’t have any Two ways to be incorrect: Say she does when she doesn’t (false alarm) Say she does not have any when she does (miss) Which is worse? Decision made by experimenter Reject H o Do not Reject H o True H o False H o You are right! Correct decision You are wrong! Type I error (false alarm) You are right! Correct decision You are wrong! Type II error (miss)

30. . Type I or type II error Does advertising affect sales? Type I error: Rejecting a true null hypothesis Saying the advertising would help sales, when it really wouldn’t help people (false alarm) Type II error: Not rejecting a false null hypothesis Saying the advertising would not help when in fact it would (miss) What would null hypothesis be? This new advertising has no effect on sales Two ways to be correct: Say it helps when it does Say it does not help when it doesn’t help Two ways to be incorrect: Say it helps when it doesn’t Say it does not help when it does Which is worse? Decision made by experimenter Reject H o Do not Reject H o True H o False H o You are right! Correct decision You are wrong! Type I error (false alarm) You are right! Correct decision You are wrong! Type II error (miss)

31. . What is worse a type I or type II error? What if we were looking at a new HIV drug that had no unpleasant side affects Type I error: Rejecting a true null hypothesis Saying the drug would help people, when it really wouldn’t help people (false alarm) Type II error: Not rejecting a false null hypothesis Saying the drug would not help when in fact it would (miss) What would null hypothesis be? This new drug has no effect on HIV Two ways to be correct: Say it helps when it does Say it does not help when it doesn’t help Two ways to be incorrect: Say it helps when it doesn’t Say it does not help when it does Which is worse? Decision made by experimenter Reject H o Do not Reject H o True H o False H o You are right! Correct decision You are wrong! Type I error (false alarm) You are right! Correct decision You are wrong! Type II error (miss)

32. . Type I or type II error What if we were looking to see if there is a fire burning in an apartment building full of cute puppies Type I error: Rejecting a true null hypothesis (false alarm) Type II error: Not rejecting a false null hypothesis (miss) What would null hypothesis be? No fire is occurring Two ways to be correct: Say “fire” when it’s really there Say “no fire” when there isn’t one Two ways to be incorrect: Say “fire” when there’s no fire (false alarm) Say “no fire” when there is one (miss) Which is worse?

33. . Type I or type II error What if we were looking to see if an individual were guilty of a crime? Type I error: Rejecting a true null hypothesis Saying the person is guilty when they are not (false alarm) Sending an innocent person to jail (& guilty person to stays free) Type II error: Not rejecting a false null hypothesis Saying the person in innocent when they are guilty (miss) Allowing a guilty person to stay free What would null hypothesis be? This person is innocent - there is no crime here Two ways to be correct: Say they are guilty when they are guilty Say they are not guilty when they are innocent Two ways to be incorrect: Say they are guilty when they are not Say they are not guilty when they are Which is worse?

34. . people taking drug people not taking drug people taking drug people not taking drug Null Hypothesis Null is TRUE Null is FALSE No effect of drug Nothing going on Drug does have effect Something going on The null hypothesis is typically that something is not present, that there is no effect, that there is no difference between population and sample or between treatment and control. There is no difference between the groups There is a difference between the groups (There are two distributions here, they are just on top of each other) (overlapping) A measure of sickness

35. . critical stat Score should fall in this region Score should fall in one of these regions Null is TRUE Null is FALSE No effect of drug Nothing going on Drug does have effect Something going on Score should fall in one of these regions people taking drug people not taking drug people taking drug people not taking drug Null is TRUE Null is FALSE A measure of sickness (There are two distributions here, they are just on top of each other) (overlapping) Remember: “procedure” vs “TRUTH”

36. 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

37. 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

38. z score = 1.64 One versus two tail test of significance: Comparing different critical scores (but same alpha level – e.g. alpha = 5%) One versus two tailed test of significance One-tailed test: If your hypothesis is “directional” claiming that one group will have a bigger mean than the other group Two-tailed test: If your hypothesis is “non-directional” claiming only that the two groups have different means How would the critical z change? Pros and cons… 5% 95% 2.5% 95% 2.5%

39. One versus two tail test of significance 5% versus 1% alpha levels -1.64 or +1.64 How would the critical z change? One-tailedTwo-tailed α = 0.05 Significance level =.05 α = 0.01 Significance level =.01 -1.96 or +1.96 -2.33 or +2.33 -2.58 or +2.58 5% 2.5% 1%.5%

40. -1.64 or +1.64 How would the critical z change? One-tailedTwo-tailed α = 0.05 Significance level =.05 α = 0.01 Significance level =.01 -1.96 or +1.96 -2.33 or +2.33 -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 One versus two tail test of significance 5% versus 1% alpha levels

41. -1.64 or +1.64 How would the critical z change? One-tailedTwo-tailed α = 0.05 Significance level =.05 α = 0.01 Significance level =.01 -1.96 or +1.96 -2.33 or +2.33 -2.58 or +2.58 What if our observed z = 1.75? Reject the null Do not Reject the null Do not Reject the null Remember, reject the null if the observed z is bigger than the critical z One versus two tail test of significance 5% versus 1% alpha levels

42. -1.64 or +1.64 How would the critical z change? One-tailedTwo-tailed α = 0.05 Significance level =.05 α = 0.01 Significance level =.01 -1.96 or +1.96 -2.33 or +2.33 -2.58 or +2.58 What if our observed z = 2.45? Reject the null Do not Reject the null Remember, reject the null if the observed z is bigger than the critical z One versus two tail test of significance 5% versus 1% alpha levels

43. Standard deviation and Variance For Sample and Population These would be helpful to know by heart – please memorize these formula Review

44. Five steps to hypothesis testing Step 1: Identify the research problem (hypothesis) Describe the null and alternative hypotheses Step 2: Decision rule Alpha level? ( α =.05 or.01)? Step 3: Calculations Step 4: Make decision whether or not to reject null hypothesis If observed z (or t) is bigger then critical z (or t) then reject null Step 5: Conclusion - tie findings back in to research problem One or two tailed test? Balance between Type I versus Type II error Critical statistic (e.g. z or t or F or r) value?

45. Summary Template There are three parts to the summary (below) The mean response time for following the sheriff’s new plan was 24 minutes, while the mean response time prior to the new plan was 30 minutes. A t-test was completed and there appears to be no significant difference in the response time following the implementation of the new plan t(9) = -1.71; n.s. Type of test with degrees of freedom Value of observed statistic n.s. = “not significant” p<0.05 = “significant” n.s. = “not significant” p<0.05 = “significant” Start summary with two means (based on DV) for two levels of the IV Describe type of test (t-test versus anova) with brief overview of results Finish with statistical summary t(4) = 1.96; ns Or if it *were* significant: t(9) = 3.93; p < 0.05

46. Degrees of Freedom Degrees of Freedom ( d.f. ) is a parameter based on the sample size that is used to determine the value of the t statistic. Degrees of freedom tell how many observations are used to calculate s, less the number of intermediate estimates used in the calculation. We lose one degree of freedom for every parameter we estimate

47. .. A note on z scores, and t score: Difference between means Variability of curve(s) Difference between means Numerator is always distance between means (how far away the distributions are or “effect size”) Denominator is always measure of variability (how wide or much overlap there is between distributions) Variability of curve(s) (within group variability)

48. . A note on variability versus effect size Difference between means Variability of curve(s) Variability of curve(s) (within group variability) Difference between means

49. .. Variability of curve(s) Variability of curve(s) (within group variability) Difference between means A note on variability versus effect size

50. . Effect size is considered relative to variability of distributions 1. Larger variance harder to find significant difference Treatment Effect Treatment Effect 2. Smaller variance easier to find significant difference x x

51. . Effect size is considered relative to variability of distributions Treatment Effect Treatment Effect x x Variability of curve(s) (within group variability) Difference between means

52. Five steps to hypothesis testing Step 1: Identify the research problem (hypothesis) Describe the null and alternative hypotheses Step 2: Decision rule: find “critical z” score Alpha level? ( α =.05 or.01)? Step 3: Calculations Step 4: Make decision whether or not to reject null hypothesis If observed z (or t) is bigger then critical z (or t) then reject null Step 5: Conclusion - tie findings back in to research problem Population versus sample standard deviation How is a t score different than a z score? One versus two-tailed test

53. Comparing z score distributions with t-score distributions Similarities include: Using bell-shaped distributions to make confidence interval estimations and decisions in hypothesis testing Use table to find areas under the curve (different table, though – areas often differ from z scores) z-scores t-scores Summary of 2 main differences: We are now estimating standard deviation from the sample (We don’t know population standard deviation) We have to deal with degrees of freedom

54. Comparing z score distributions with t-score distributions 2) The shape of the sampling distribution is very sensitive to small sample sizes (it actually changes shape depending on n) Please notice: as sample sizes get smaller, the tails get thicker. As sample sizes get bigger tails get thinner and look more like the z-distribution Differences include: 1)We use t-distribution when we don’t know standard deviation of population, and have to estimate it from our sample

55. Comparing z score distributions with t-score distributions Differences include: 1)We use t-distribution when we don’t know standard deviation of population, and have to estimate it from our sample Critical t (just like critical z) separates common from rare scores Critical t used to define both common scores “confidence interval” and rare scores “region of rejection”

56. Comparing z score distributions with t-score distributions 2) The shape of the sampling distribution is very sensitive to small sample sizes (it actually changes shape depending on n) Differences include: 1)We use t-distribution when we don’t know standard deviation of population, and have to estimate it from our sample Please notice: as sample sizes get smaller, the tails get thicker. As sample sizes get bigger tails get thinner and look more like the z-distribution

57. Comparing z score distributions with t-score distributions 2) The shape of the sampling distribution is very sensitive to small sample sizes (it actually changes shape depending on n) Differences include: 1)We use t-distribution when we don’t know standard deviation of population, and have to estimate it from our sample 3) Because the shape changes, the relationship between the scores and proportions under the curve change (So, we would have a different table for all the different possible n’s but just the important ones are summarized in our t-table) Please note: Once sample sizes get big enough the t distribution (curve) starts to look exactly like the z distribution (curve) scores

58. A quick re-visit with the law of large numbers Relationship between increased sample size decreased variability smaller “critical values” As n goes up variability goes down

59. Law of large numbers: As the number of measurements increases the data becomes more stable and a better approximation of the true signal (e.g. mean) As the number of observations (n) increases or the number of times the experiment is performed, the signal will become more clear (static cancels out) http://www.youtube.com/watch?v=ne6tB2KiZuk With only a few people any little error is noticed (becomes exaggerated when we look at whole group) With many people any little error is corrected (becomes minimized when we look at whole group)

60. . Interpreting t-table Technically, we have a different t-distribution for each sample size This t-table summarizes the most useful values for several distributions n = 17 n = 5 This t-table presents useful values for distributions (organized by degrees of freedom) 1.962.58 1.64 Remember these useful values for z-scores? We use degrees of freedom (df) to approximate sample size Each curve is based on its own degrees of freedom (df) - based on sample size, and its own table tying together t-scores with area under the curve

61. Area between two scores Area beyond two scores (out in tails) Area beyond two scores (out in tails) Area in each tail (out in tails) Area in each tail (out in tails) df

62. Area between two scores Area beyond two scores (out in tails) Area beyond two scores (out in tails) Area in each tail (out in tails) Area in each tail (out in tails) Notice with large sample size it is same values as z-score. 1.962.58 1.64 Remember these useful values for z-scores? df

63. Degrees of Freedom Degrees of Freedom ( d.f. ) is a parameter based on the sample size that is used to determine the value of the t statistic. Degrees of freedom tell how many observations are used to calculate s, less the number of intermediate estimates used in the calculation.

64. Five steps to hypothesis testing Step 1: Identify the research problem (hypothesis) Describe the null and alternative hypotheses Step 2: Decision rule Alpha level? ( α =.05 or.01)? Step 3: Calculations Step 4: Make decision whether or not to reject null hypothesis If observed z (or t) is bigger then critical z (or t) then reject null Step 5: Conclusion - tie findings back in to research problem One or two tailed test? Balance between Type I versus Type II error Critical statistic (e.g. z or t or F or r) value?

65. In-class writing assignment