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Please start portfolios. Null Hypothesis. 1.  Question: Are men and women the same height Null: There is no difference in the height of men and women 2.  Question: Is marijuana or beer more dangerous

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  1. Please start portfolios

  2. Null Hypothesis • 1.  Question: Are men and women the same height • Null: There is no difference in the height of men and women • 2. Question: Is marijuana or beer more dangerous • Null: There is no difference between marijuana and beer in terms of how dangerous they are • 3. Question: Does advertising affect sales • Null: Advertising has no effect on sales • 4. Question: Does the sleep medicine change the number of hours you sleep at night? • Null: Sleep medicine does not affect how much you sleep • 5. Question: Is the building on fire • Null: The building is not on fire • 6. Question: Is he guilty • Null: He is not guilty

  3. Writing Assignment • 1.  What are the three propositions of the Central Limit Theorem • 2.  What is a null hypothesis? • 3.  How do you know when to “reject the null hypothesis” • 4.  What are three synonyms for “reject the null hypothesis” • 5.  Write the formula for • Standard deviation of the sample • Standard deviation of the population • Variance of the sample • Variance of the population

  4. MGMT 276: Statistical Inference in ManagementMcClelland Hall, Room 1328:30 – 10:45 Monday - ThursdaySummer II, 2012. Welcome Experiment

  5. Please read: Chapters 5 - 9 in Lind book & Chapters 10, 11, 12 & 14 in Plous book: Lind Chapter 5: Survey of Probability Concepts Chapter 6: Discrete Probability Distributions Chapter 7: Continuous Probability Distributions Chapter 8: Sampling Methods and CLT Chapter 9: Estimation and Confidence Interval Plous Chapter 10: The Representativeness Heuristic Chapter 11: The Availability Heuristic Chapter 12: Probability and Risk Chapter 14: The Perception of Randomness

  6. Use this as your study guide By the end of lecture today7/23/12 Logic of hypothesis testing Steps for hypothesis testing Levels of significance (Levels of alpha) what does alpha of .05 mean? what does p < 0.05 mean? what does alpha of .01 mean? what does p < 0.01 mean? Type I vs Type II Error

  7. Rejecting the null hypothesis . notnull null big z score x x • 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 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 . null x x small z score

  8. 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 x2) • observed stat > critical stat If we want to reject the null, we want our t (or z or r or F or x2) 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

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

  10. Writing Assignment As n ↑ xwill approach µ • 1.  What are the three propositions of the Central Limit Theorem • 2.  What is a null hypothesis? • 3.  How do you know when to “reject the null hypothesis” • 4.  What are three synonyms for “reject the null hypothesis” • 5.  Write the formula for • Standard deviation of the sample • Standard deviation of the population • Variance of the sample • Variance of the population s s As n ↑ curve will approach normal shape Central Limit Theorem – 3 propositions As n ↑ curve variability gets smaller SEM (sample) SEM

  11. Writing Assignment • 1.  What are the three propositions of the Central Limit Theorem • 2.  What is a null hypothesis? • The null hypothesis states that nothing unusual is happening • “There is no difference between groups”

  12. Writing Assignment 1.  What are the three propositions of the Central Limit Theorem 2.  What is a null hypothesis? 3.  How do you know when to “reject the null hypothesis” 4.  What are three synonyms for “reject 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 • the result is “statistically significant” • the observed statistic is larger than the critical statistic (which can be a ‘z” or “t” or “r” or “F” or x2) • observed stat > critical stat • the p value is less than 0.05 (which is our alpha) • p < 0.05 • we reject the null hypothesis • then we have support for our alternative hypothesis

  13. Writing Assignment • 1.  What are the three propositions of the Central Limit Theorem • 2.  What is a null hypothesis? • 3.  How do you know when to “reject the null hypothesis” • 4.  What are three synonyms for “reject the null hypothesis” • 5.  Write the formula for • Standard deviation of the sample • Standard deviation of the population • Variance of the sample • Variance of the population

  14. . Where are we? As n ↑ curve will approach normal shape Central Limit Theorem – 3 propositions As n ↑ xwill approach µ s As n ↑ curve variability gets smaller s SEM (sample) SEM Normal Curve • Notice: • 3 types of numbers • raw scores • z scores • probabilities outliers not typical score not typical score typical score Critical z used to define both common scores “confidence interval” And rare scores “region of rejection”

  15. Measurements that occur within the middle part of the curve are ordinary (typical) and probably belong there For scores that fall into the middle range, we do not reject the null Moving from descriptive stats into inferential stats…. Critical z 1.64 Critical z -1.64 90% 5% 5% Measurements that occur outside this middle ranges are suspicious, may be an error or belong elsewhere For scores that fall into the regions of rejection, we reject the null What percent of the distribution will fall in region of rejection Critical Values http://today.msnbc.msn.com/id/33411196/ns/today-today_health/ http://www.youtube.com/watch?v=0r7NXEWpheg

  16. 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 x2) 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”

  17. . 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?

  18. . 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. Null Hypothesis A measure of sickness people taking drugpeople not taking drug (There are two distributions here, they are just on top of each other) (overlapping) people taking drug people not taking drug A measure of sickness A measure of sickness Null is FALSE Null is TRUE Drug does have effect Something going on Nothing going on No effect of drug There is no difference between the groups There is a difference between the groups

  19. Remember: “procedure” vs “TRUTH” . (There are two distributions here, they are just on top of each other) (overlapping) A measure of sickness people taking drug people not taking drug people taking drugpeople not taking drug A measure of sickness A measure of sickness Null is FALSE Null is TRUE Score should fall in this region critical stat critical stat critical stat critical stat Score should fall in one of these regions 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

  20. . Two ways to be right: Status of Null Hypothesis(actually, via magic truth-line) True Ho False Ho Do notReject Ho Decision madeby experimenter Reject Ho 1. “Reject a false null hypothesis” “there really is something going on” 2. “Do not reject a true null hypothesis” “there really is no difference between groups” You are right! Correct decision You are right! Correct decision

  21. . Two ways to be wrong: Status of Null Hypothesis(actually, via magic truth-line) True Ho False Ho Do notReject Ho Decision madeby experimenter Reject Ho 1. “Reject a true null hypothesis” say there’s a difference when there’s not (Type I)The score fell in the tails but the null was actually “TRUE” 2. “Do not reject a false null hypothesis” say there really is no difference between groupswhen there really is (Type II) The score fell in the middle but the null was still “FALSE” You are wrong! Type II error(miss) You are wrong! Type I error(false alarm)

  22. Possible outcomes of hypothesis test Status of Null Hypothesis(actually, via magic truth-line) True Ho False Ho Do notReject Ho Decision madeby experimenter Reject Ho You are wrong! Type II error(miss) You are right! Correct decision You are wrong! Type I error(false alarm) You are right! Correct decision • Probability of rejecting a true null hypothesis = alpha • The alpha you choose becomes the probability of • making a Type I error

  23. . What’s worse – Type I or Type II error? Status of Null Hypothesis(actually, via magic truth-line) True Ho False Ho Do notReject Ho Decision madeby experimenter Reject Ho . You are wrong! Type II error(miss) You are right! Correct decision You are wrong! Type I error(false alarm) You are right! Correct decision

  24. We make decisions at Security Check Points . .

  25. . Type I or Type II error? . Does this airline passengerhave 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?

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

  27. . Type I or type II error? True Ho False Ho You are right! Correct decision You are wrong! Type II error(miss) Do notReject Ho Decision madeby experimenter You are wrong! Type I error(false alarm) You are right! Correct decision Reject Ho Does this airline passenger have a 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? What would null hypothesis be? This passenger does not have any 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)

  28. . Type I or type II error True Ho False Ho You are right! Correct decision You are wrong! Type II error(miss) Do notReject Ho Decision madeby experimenter You are wrong! Type I error(false alarm) You are right! Correct decision Reject Ho Does advertising affect sales? • Two ways to be correct: • Say it helps when it does • Say it does not help when it doesn’t help Which is worse? • Two ways to be incorrect: • Say it helps when it doesn’t • Say it does not help when it does What would null hypothesis be? This new advertising has no effect on 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)

  29. . What is worse a type I or type II error? True Ho False Ho You are right! Correct decision You are wrong! Type II error(miss) Do notReject Ho Decision madeby experimenter You are wrong! Type I error(false alarm) You are right! Correct decision Reject Ho What if we were lookingat a new HIV drug that had no unpleasant side affects • 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? What would null hypothesis be? This new drug has no effect on HIV 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)

  30. . Type I or type II error Which is worse? What if we were looking to see if there is a fire burning in an apartment building full of cute puppies • 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) What would null hypothesis be? No fire is occurring Type I error: Rejecting a true null hypothesis (false alarm) Type II error: Not rejecting a false null hypothesis (miss)

  31. . Type I or type II error Which is worse? What if we were looking to see if an individualwere guilty of a crime? • 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 What would null hypothesis be? This person is innocent - there is no crime here 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

  32. . . Which is worse? A. Sending an innocent man to prison (Type I error – false alarm) • Letting a guilty man go free • (Type II error – miss)

  33. . What is worse a type I or type II error?Another example Which is worse? . What if we were looking to see if our management program works better than old program What would null hypothesis be? What would rejecting the null be? What would accepting the null be? Type I error: Saying that…(false alarm) Type II error: Saying that… (miss)

  34. Generate an example of a situation that requires a decision to be made • For each situation • identify the null hypothesis • and describe how you would decide to • reject null correctly • reject null erroneously (false alarm - type I error) • not reject null correctly • not reject null incorrectly (miss - type II error) Remember there are two parts: Conclusion – when in fact – Truth Example of Type I Error We concluded he was guilty (reject the null) When in fact He actually was innocent

  35. Status of Null Hypothesis(actually, via magic truth-line) True Ho False Ho Do notReject Ho Decision madeby experimenter Reject Ho Worksheet Type I versus Type II Error You are wrong! Type II error(miss) You are right! Correct decision You are wrong! Type I error(false alarm) You are right! Correct decision • Probability of rejecting a true null hypothesis = alpha • The alpha you choose becomes the probability of • making a Type I error

  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 x2) • observed stat > critical stat If we want to reject the null, we want our t (or z or r or F or x2) 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 95%Has and alpha of 5%α = .05 Critical z 2.58 Critical z -2.58 Confidence Interval of 99% Has and alpha of 1% α = .01 99% Area in the tails is called alpha Critical z 1.96 Critical z -1.96 95% Critical Z separates rare from common scores 90% Critical z 1.64 Critical z -1.64 Confidence Interval of 90% Has and alpha of 10% α = . 10

  38. 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 95% 95% 2.5% 5% 2.5% How would the critical z change? Pros and cons…

  39. One versus two tail test of significance5% versus 1% alpha levels How would the critical z change? 2.5% .5% 5% 2.5% 1% .5% -1.64 or +1.64 -1.96 or +1.96 -2.33 or +2.33 -2.58 or +2.58

  40. One versus two tail test of significance5% versus 1% alpha levels What if our observed z = 2.0? How would the critical z change? -1.64 or +1.64 -1.96 or +1.96 Remember, reject the null if the observed z is bigger than the critical z Reject the null Reject the null -2.33 or +2.33 -2.58 or +2.58 Do notReject the null Do notReject the null

  41. One versus two tail test of significance5% versus 1% alpha levels What if our observed z = 1.75? How would the critical z change? -1.64 or +1.64 -1.96 or +1.96 Do not Reject the null Remember, reject the null if the observed z is bigger than the critical z Reject the null -2.33 or +2.33 -2.58 or +2.58 Do notReject the null Do notReject the null

  42. One versus two tail test of significance5% versus 1% alpha levels What if our observed z = 2.45? How would the critical z change? -1.64 or +1.64 -1.96 or +1.96 Remember, reject the null if the observed z is bigger than the critical z Reject the null Reject the null -2.33 or +2.33 -2.58 or +2.58 Do notReject the null Reject the null

  43. . . A note on z scores, and t score: • 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) Difference between means Difference between means Variability of curve(s)(within group variability) Variabilityof curve(s)

  44. Five steps to hypothesis testing Step 1: Identify the research problem (hypothesis) Describe the null and alternative hypotheses How is a t score same as a z score? How is a t score different than a z score? Step 2: Decision rule • Alpha level? (α= .05 or .01)? • Critical statistic (e.g. z or t) value? 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 Population versus sample standard deviation Population versus sample standard deviation Step 5: Conclusion - tie findings back in to research problem

  45. Thank you! See you next time!!

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