MGMT 276: Statistical Inference in Management.

1. MGMT 276: Statistical Inference in Management. Welcome http://www.thedailyshow.com/video/index.jhtml?videoId=188474&title=an-arab-family-man

2. 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

3. Use this as your study guide By the end of lecture today2/24/11 Probability of an event Complement of an event; Union of two events Intersection of two events; Mutually exclusive events Collectively exhaustive events Conditional probability Law of Large Numbers Central Limit Theorem

4. Please click in Homework due next class - (Due March 1st) My last name starts with a letter somewhere between A. A – D B. E – L C. M – R D. S – Z Complete Gilbert Reading Available on class website Please double check – All cell phones other electronic devices are turned off and stowed away Turn your clicker on

5. Two mutually exclusive characteristics: if the occurrence of any one of them automatically implies the non-occurrence of the remaining characteristic Two events are mutually exclusive if they cannot occur at the same time (i.e. they have no outcomes in common). Two propositions that logically cannot both be true. NoWarranty Warranty For example, a car repair is either covered by the warranty (A) or not (B). http://www.thedailyshow.com/video/index.jhtml?videoId=188474&title=an-arab-family-man

6. Collectively Exhaustive Events Events are collectively exhaustive if their union isthe entire sample space S. Two mutually exclusive, collectively exhaustive events are dichotomous (or binary) events. For example, a car repair is either covered by the warranty (A) or not (B). NoWarranty Warranty

7. NoWarranty Satirical take on being “mutually exclusive” Warranty Recently a public figure in the heat of the moment inadvertently made a statement that reflected extreme stereotyping that many would find highly offensive. It is within this context that comical satirists have used the concept of being “mutually exclusive” to have fun with the statement. Decent , family man Arab Transcript: Speaker 1: “He’s an Arab” Speaker 2: “No ma’am, no ma’am. He’s a decent, family man, citizen…” http://www.thedailyshow.com/video/index.jhtml?videoId=188474&title=an-arab-family-man

8. Union versus Intersection ∩ P(A B) Union of two events means Event A or Event B will happen Intersection of two events means Event A and Event B will happen Also called a “joint probability” P(A ∩ B)

9. The union of two events: all outcomes in the sample space S that are contained either in event Aor in event Bor both (denoted A  B or “A or B”).  may be read as “or” since one or the other or both events may occur.

10. The union of two events: all outcomes contained either in event Aor in event Bor both (denoted A  B or “A or B”). What is probability of drawing a red card or a queen? what is Q  R? It is the possibility of drawing either a queen (4 ways) or a red card (26 ways) or both (2 ways).

11. Probability of picking a Queen Probability of picking a Red 26/52 4/52 P(Q) = 4/52(4 queens in a deck) 2/52 P(R) = 26/52 (26 red cards in a deck) P(Q  R) = 2/52 (2 red queens in a deck) Probability of picking both R and Q When you add the P(A) and P(B) together, you count the P(A and B) twice. So, you have to subtract P(A  B) to avoid over-stating the probability. P(Q  R) = P(Q) + P(R) – P(Q  R) = 4/52 + 26/52 – 2/52 = 28/52 = .5385 or 53.85%

12. Union versus Intersection ∩ P(A B) Union of two events means Event A or Event B will happen Intersection of two events means Event A and Event B will happen Also called a “joint probability” P(A ∩ B)

13. The intersection of two events: all outcomes contained in both event A and event B(denoted A  B or “A and B”) What is probability of drawing red queen? what is Q R? It is the possibility of drawing both a queen and a red card (2 ways).

14. If two events are mutually exclusive (or disjoint) their intersection is a null set (and we can use the “Special Law of Addition”) P(A ∩ B) = 0 Intersection of two events means Event A and Event B will happen Examples: mutually exclusive If A = Poodles If B = Labradors Poodles and Labs:Mutually Exclusive (assuming purebred)

15. If two events are mutually exclusive (or disjoint) their intersection is a null set (and we can use the “Special Law of Addition”) P(A ∩ B) = 0 ∩ Dog Pound P(A B) = P(A) +P(B) Intersection of two events means Event A and Event B will happen Examples: If A = Poodles If B = Labradors (let’s say 10% of dogs are poodles) (let’s say 15% of dogs are labs) What’s the probability of picking a poodle or a lab at random from pound? P(poodle or lab) = P(poodle) + P(lab) P(poodle or lab) = (.10) + (.15) = (.25) Poodles and Labs:Mutually Exclusive (assuming purebred)

16. Conditional Probabilities Probability that A has occurred given that B has occurred Denoted P(A | B): The vertical line “ | ” is read as “given.” P(A ∩ B) P(A | B) = P(B) The sample space is restricted to B, an event that has occurred. A  B is the part of B that is also in A. The ratio of the relative size of A  B to B is P(A | B).

17. Conditional Probabilities Probability that A has occurred given that B has occurred Of the population aged 16 – 21 and not in college: P(U) = .1350 P(ND) = .2905 P(UND) = .0532 What is the conditional probability that a member of this population is unemployed, given that the person has no diploma? .0532 P(A ∩ B) .1831 = P(A | B) = = .2905 P(B) or 18.31%

18. Conditional Probabilities Probability that A has occurred given that B has occurred Of the population aged 16 – 21 and not in college: P(U) = .1350 P(ND) = .2905 P(UND) = .0532 What is the conditional probability that a member of this population is unemployed, given that the person has no diploma? .0532 P(A ∩ B) .1831 = P(A | B) = = .2905 P(B) or 18.31%

19. Law of large numbers: As the number of measurements increases the data becomes more stable and a better approximation of the true (theoretical) probability As the number of observations (n) increases or the number of times the experiment is performed, the estimate will become more accurate.

20. Sampling distributions of sample means versus frequency distributions of individual scores Distribution of raw scores: is an empirical probability distribution of the values from a sample of raw scores from a population Eugene X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X • Frequency distributions of individual scores • derived empirically • we are plotting raw data • this is a single sample Melvin X X X X X X X X X X X X Take a single score x Repeat over and over x x x Population x x x x

21. Sampling distribution: is a theoretical probability distribution of • the possible values of some sample statistic that would • occur if we were to draw an infinite number of same-sized • samples from a population important note: “fixed n” • Sampling distributions of sample means • theoretical distribution • we are plotting means of samples Take sample – get mean Repeat over and over Population

22. Sampling distribution: is a theoretical probability distribution of • the possible values of some sample statistic that would • occur if we were to draw an infinite number of same-sized • samples from a population important note: “fixed n” • Sampling distributions of sample means • theoretical distribution • we are plotting means of samples Take sample – get mean Repeat over and over Population Distribution of means of samples

23. Sampling distribution: is a theoretical probability distribution of • the possible values of some sample statistic that would • occur if we were to draw an infinite number of same-sized • samples from a population Eugene • Frequency distributions of individual scores • derived empirically • we are plotting raw data • this is a single sample X X X X X X X Melvin X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X • Sampling distributions sample means • theoretical distribution • we are plotting means of samples 23rd sample 2nd sample

24. X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X Sampling distribution for continuous distributions • Central Limit Theorem: If random samples of a fixed N are drawn • from any population (regardless of the shape of the • population distribution), as N becomes larger, the • distribution of sample means approaches normality, with • the overall mean approaching the theoretical population • mean. Distribution of Raw Scores Sampling Distribution of Sample means Melvin 23rd sample Eugene X X X X X 2nd sample

25. Sampling distributions of sample means versus frequency distributions of individual scores • Sampling distribution: is a theoretical probability distribution of • the possible values of some sample statistic that would • occur if we were to draw an infinite number of same-sized • samples from a population Eugene X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X An example of frequency distributions for population of individual scores Melvin X X X X X X X X µ= 100 X X Mean = 100 X X σ= 3 100 Notice: SEM is smaller than SD – especially as n increases Standard Deviation = 3 23rd sample An example of a sampling distribution of sample means 2nd sample µ = 100 Mean = 100 = 1 Standard Error of the Mean = 1 100

26. Sampling distributions of sample means versus frequency distributions of individual scores • Sampling distribution: is a theoretical probability distribution of • the possible values of some sample statistic that would • occur if we were to draw an infinite number of same-sized • samples from a population Eugene X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X An example of frequency distributions for population of individual scores Melvin X X X X X X X X µ= 100 X X Mean = 100 X X σ= 3 100 Notice: SEM is smaller than SD – especially as n increases Standard Deviation = 3 23rd sample An example of a sampling distribution of sample means 2nd sample µ = 100 Mean = 100 = 1 Standard Error of the Mean = 1 100

27. Sampling distribution: is a theoretical probability distribution of • the possible values of some sample statistic that would • occur if we were to draw an infinite number of same-sized • samples from a population Notice: SEM is smaller than SD – especially as n increases X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X µ= 100 X X Mean = 100 X X σ= 3 100 Standard Deviation = 3 An example of a sampling distribution of sample means µ = 100 Mean = 100 = 1 Standard Error of the Mean = 1 100

28. X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X Sampling distribution for continuous distributions • Central Limit Theorem: If random samples of a fixed N are drawn • from any population (regardless of the shape of the • population distribution), as N becomes larger, the • distribution of sample means approaches normality, with • the overall mean approaching the theoretical population • mean. Distribution of Raw Scores Sampling Distribution of Sample means Melvin Eugene X X X X X

29. Sampling distribution: is a theoretical probability distribution of • the possible values of some sample statistic that would • occur if we were to draw an infinite number of same-sized • samples from a population Eugene • Frequency distributions of individual scores • derived empirically • we are plotting raw data • this is a single sample X X X X X X X Melvin X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X • Sampling distributions sample means • theoretical distribution • we are plotting means of samples 23rd sample 2nd sample Important

30. Sampling distributions of sample means versus frequency distributions of individual scores • Sampling distribution: is a theoretical probability distribution of • the possible values of some sample statistic that would • occur if we were to draw an infinite number of same-sized • samples from a population In principle, sampling distributions exist for means, standard deviations, proportions and correlations (among others)

31. Central Limit Theorem Proposition 1: If sample size (n) is large enough (e.g. 100) The mean of the sampling distribution will approach the mean of the population As n ↑ x will approach µ Proposition 2: If sample size (n) is large enough (e.g. 100) The sampling distribution of means will be approximately normal, regardless of the shape of the population As n ↑ curve will approach normal shape Proposition 3: The standard deviation of the sampling distribution equals the standard deviation of the population divided by the square root of the sample size. As n increases SEM decreases. As n ↑ curve variability gets smaller X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X

32. Central Limit Theorem Proposition 1: If sample size (n) is large enough (e.g. 100) The mean of the sampling distribution will approach the mean of the population Law of large numbers: As the number of measurements increases the data becomes more stable and a better approximation of the true (theoretical) probability. Larger sample sizes tend to be associated with stability. As the number of observations (n) increases or the number of times the experiment is performed, the estimate will become more accurate.

33. Take sample (n = 5) – get mean Proposition 2: If sample size (n) is large enough (e.g. 100), the sampling distribution of means will be approximately normal, regardless of the shape of the population Repeat over and over Population population population population sampling distribution n = 2 sampling distribution n = 5 sampling distribution n = 4 sampling distribution n = 30 sampling distribution n = 5 sampling distribution n = 25

34. Central Limit Theorem Proposition 2: If sample size (n) is large enough (e.g. 100) The sampling distribution of means will be approximately normal, regardless of the shape of the population

35. Central Limit Theorem Proposition 2: If sample size (n) is large enough (e.g. 100) The sampling distribution of means will be approximately normal, regardless of the shape of the population

36. Central Limit Theorem Proposition 2: If sample size (n) is large enough (e.g. 100) The sampling distribution of means will be approximately normal, regardless of the shape of the population

37. Central Limit Theorem Proposition 2: If sample size (n) is large enough (e.g. 100) The sampling distribution of means will be approximately normal, regardless of the shape of the population • If sample size (n) is not large (but the shape of the • population is normal), the sampling distribution of • means will be normal. X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X

38. Central Limit Theorem Proposition 3: The standard deviation of the sampling distribution equals the standard deviation of the population divided by the square root of the sample size. As n increases SEM decreases. X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X

39. Central Limit Theorem Proposition 1: If sample size (n) is large enough (e.g. 100) The mean of the sampling distribution will approach the mean of the population As n ↑ x will approach µ Proposition 2: If sample size (n) is large enough (e.g. 100) The sampling distribution of means will be approximately normal, regardless of the shape of the population As n ↑ curve will approach normal shape Proposition 3: The standard deviation of the sampling distribution equals the standard deviation of the population divided by the square root of the sample size. As n increases SEM decreases. As n ↑ curve variability gets smaller X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X

40. X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X Sampling distribution of sample means • Central Limit Theorem: If random samples of a fixed N are drawn • from any population (regardless of the shape of the • population distribution), as N becomes larger, the • distribution of sample means approaches normality, with • the overall mean approaching the theoretical population • mean. Distribution of Raw Scores Sampling Distribution of Sample means Melvin Eugene X X X X X

41. Central Limit Theorem: If random samples of a fixed N are drawn from any population (regardless of the shape of the population distribution), as N becomes larger, the distribution of sample means approaches normality, with the overall mean approaching the theoretical population mean. Distribution of Raw Scores Animation for creating sampling distribution of sample means Distribution of single sample Eugene Melvin Sampling Distribution of Sample means Sampling Distribution of Sample means Mean for sample 12 • Mean for sample 7 http://onlinestatbook.com/stat_sim/sampling_dist/index.html

42. X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X Sampling distribution for continuous distributions • Central Limit Theorem: If random samples of a fixed N are drawn • from any population (regardless of the shape of the • population distribution), as N becomes larger, the • distribution of sample means approaches normality, with • the overall mean approaching the theoretical population • mean. Distribution of Raw Scores Sampling Distribution of Sample means Melvin 23rd sample Eugene X X X X X 2nd sample

43. Central Limit Theorem Proposition 1: If sample size (n) is large enough (e.g. 100) The mean of the sampling distribution will approach the mean of the population As n ↑ x will approach µ Proposition 2: If sample size (n) is large enough (e.g. 100) The sampling distribution of means will be approximately normal, regardless of the shape of the population As n ↑ curve will approach normal shape Proposition 3: The standard deviation of the sampling distribution equals the standard deviation of the population divided by the square root of the sample size. As n increases SEM decreases. As n ↑ curve variability gets smaller X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X

44. . Writing Assignment: Writing a letter to a friend • Imagine you have a good friend (pick one). This is a good friend whom you consider to be smart and interested in stuff generally. They are teaching themselves stats (hoping to test out of the class) but need your help on a couple ideas. For this assignment please write your friend/mom/dad/ favorite cousin a letter answering these five questions: (Feel free to use diagrams and drawings if you think that can help) • Dear Friend, • 1. I’m struggling with this whole Central Limit Theorem idea. Could you • describe for me the difference between a distribution of raw scores, and a • distribution of sample means? • 2. I also don’t get the “three propositions of the Central Limit Theorem”. They all • seem to address sample size, but I don’t get how sample size could affect • these three things. If you could help explain it, that would be really helpful.

46. Thank you! See you next time!!