MGMT 276: Statistical Inference in Management Spring, 2014

1. MGMT 276: Statistical Inference in ManagementSpring, 2014 Welcome Green sheets

2. Please click in My last name starts with a letter somewhere between A. A – D B. E – L C. M – R D. S – Z

3. Homework due – Tuesday (March 4th) • On class website: • Please print and complete homework worksheet #9 • Calculating z-score, raw scores and probabilities using the normal curve

4. Please read before our next exam (March 25th) - Chapters 5 - 11 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 Chapter 10: One sample Tests of Hypothesis Chapter 11: Two sample Tests of Hypothesis Plous Chapter 10: The Representativeness Heuristic Chapter 11: The Availability Heuristic Chapter 12: Probability and Risk Chapter 14: The Perception of Randomness We’ll be jumping around some…we will start with chapter 7

5. Raw scores, z scores & probabilities • Notice: • 3 types of numbers • raw scores • z scores • probabilities Mean = 50 Standard deviation = 10 z = -2 z = +2 If we go up two standard deviations z score = +2.0 and raw score = 70 If we go down two standard deviations z score = -2.0 and raw score = 30

6. Normal distribution Raw scores z-scores probabilities Z Scores Have z Find raw score Have z Find area z table Formula Have area Find z Area & Probability Raw Scores Have raw score Find z

7. . Homework Worksheet

8. Hint: Always draw a picture! Homework worksheet

9. . Homework Worksheet: Problem 1 1 sd 1 sd .68 30 32 28

10. . Homework Worksheet: Problem 2 2 sd 2 sd .95 32 28 34 26 30

11. . Homework Worksheet: Problem 3 3 sd 3 sd .997 24 36 32 28 34 26 30

12. . Homework Worksheet: Problem 4 .50 24 36 32 28 34 26 30

13. . Homework Worksheet: Problem 5 Go to table 33-30 z = 1.5 z = .4332 2 .4332 24 36 32 28 34 26 30

14. . Homework Worksheet: Problem 6 Go to table 33-30 z = 1.5 z = .4332 2 .9332 .4332 .5000 24 36 32 28 34 26 30

15. .0668 Go to table 33-30 .4332 z = 1.5 z = .4332 2 33 .5000 - .4332 = .0668 Go to table 29-30 z =-.5 z = .1915 .5000 .1915 2 .5000 + .1915 = .6915 29 .4938 .1915 25-30 25 31 z = -2.5 z = .4938 2 .4938 + .1915 = .6853 Go to table 31-30 z =.5 z = .1915 2 .0668 .4332 27-30 z = -1.5 z = .4332 27 .5000 - .4332 = .0668 2

16. Homework Worksheet Problem 11: .5000 + .4938 = .9938 Problem 12: .5000 - .3413 = .1587 Problem 13: 30 Problem 14: 28 and 32

17. . 77th percentile Go to table nearest z = .74 .2700 x = mean + z σ = 30 + (.74)(2) = 31.48 .7700 .27 .5000 24 36 ? 28 34 26 30 31.48

18. . 13th percentile Go to table nearest z = 1.13 .3700 x = mean + z σ = 30 + (-1.13)(2) = 27.74 Note: .13 + .37 = .50 .37 .50 .13 ? 24 36 32 27.74 34 26 30

19. Homework Worksheet Problem 17: 68% or .68 or 68.26% or .6826 Problem 18: 95% or .95 or 95.44% or .9544 Problem 19: 99.70% or .9970 Problem 20: 27.34% or .2734

20. Please use the following distribution with a mean of 200 and a standard deviation of 40. Find the area under the curve between scores of 200 and 230. Start by filling in the desired information on curve 20 (to the right)(Note this one will require you to calculate a z-score for a raw score of 230 and use the z-table) Go to table 230-200 z = .75 z = .2734 40 .2734 80 320 240 160 280 120 200

21. Homework Worksheet Problem 21: 40.13% or .4013 Problem 22: 69.15% or .6915 Problem 23: 18.41% or .1841 Problem 24: 28.81% or .2881 Problem 25: 96.93% or .9693 or 96.93% or .9693 Problem 26: .89% or .0089 Problem 27: 95.99% or .9599 Problem 28: 4.01% or .0401 Problem 29: 293.2 x = mean + z σ = 200 + (2.33)(40) = 293.2 Problem 30: 182.4 x = mean + z σ = 200 + (-.44)(40) = 182.4 Problem 31: 190 Problem 32: 217.6

22. . Find score associated with the 75th percentile 75th percentile Go to table nearest z = .67 .2500 x = mean + z σ = 30 + (.67)(2) = 31.34 .7500 .25 .5000 24 36 ? 28 34 26 30 31.34 z = .67

23. . Find the score associated with the 25th percentile 25th percentile Go to table nearest z = -.67 .2500 x = mean + z σ = 30 + (-.67)(2) = 28.66 .2500 .25 .25 28.66 24 ? 36 28 34 26 30 z = -.67

25. Variability and means Variability and means 38 40 44 48 52 56 58 The variability is different…. The mean is the same … What might the standard deviation be? 38 40 44 48 52 56 58 Remember to keep number lines same for both examples

26. Variability and means Grades of all students in the class • 65 70 75 80 85 90 • Grades Grades of “C” students What might the standard deviation be? What might this be an example of? • 65 70 75 80 85 90 • Grades Other examples?

27. Variability and means Remember, there is an implied axis measuring frequency f 60 65 70 75 80 85 90 f Remember to keep number lines equally spaced 60 65 70 75 80 85 90 Remember to keep number lines same for both examples Variable must be numeric

28. Variability and means Birth weight for infants From entire population 1 3 5 7 9 11 13 Birth weight in pounds Birth weight for infants from a “typical family” What might the standard deviation be? What might this be an example of? • 3 5 7 9 11 13 • Birth weight in pounds Other examples? Notice: number lines equally spaced

29. Variability and means Social distance norm(personal space) for international community 40 50 60 70 80 90 100Social Distance Norm Social distance norm (personal space) for Tucson What might the standard deviation be? What might this be an example of? 40 50 60 70 80 90 100 Social Distance Norm Other examples? Notice: number lines equally spaced

30. Variability and means Distributions same mean different variability Final exam scores “C” students versus whole class Birth weight within a typical family versus within the whole community Running speed 30 year olds vs. 20 – 40 year olds Number of violent crimes Milwaukee vs. whole Midwest Social distance (personal space) California vs international community

31. Variability and means Distributions different mean same variability Performance on a final exam Before versus after taking the class 40 50 60 70 80 90 100 Score on final (before taking class) 40 50 60 70 80 90 100 Score on final (before taking class) Notice: number lines equally spaced

32. Variability and means Distributions different mean same variability Height of men versus women 62 64 66 68 70 72 74 76Inches in height (women) 62 64 66 68 70 72 74 76Inches in height (men) Notice: number lines equally spaced

33. Variability and means Distributions different mean same variability Driving ability Talking on a cell phone or not 2 4 6 8 10 12 14 16Number of errors (not on phone) 2 4 6 8 10 12 14 16Number of errors (on phone) Notice: number lines equally spaced

34. Variability and means Comparing distributions different mean same variability Performance on a final exam Before versus after taking the class Height of men versus women Driving ability Talking on a cell phone or not Notice: number lines equally spaced

35. . Writing Assignment Comparing distributions (mean and variability) • Think of examples for these three situations • same mean but different variability • same variability but different means • same mean and same variability (different groups) • estimate standard deviation • calculate variance • for each curve find the raw score for the z’s given Remember: number lines equally spaced

36. Hand in homework & worksheet

37. . 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 .9500 .475 .475 26.08 33.92 ? ? 24 32 36 28 30

38. Remember confidence intervals? 95% Confidence Interval: We can be 95% confident that our population mean falls between these two scores 99% Confidence Interval: We can be 99% confident that our population mean falls between these two scores z- scores allow us to find the raw scores for the middle 95% of the distribution

39. Standard Error of the Mean (SEM) Remember confidence intervals? Revisit Confidence Intervals 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. Greater confidence implies loss of precision.(95% confidence is most often used) Can actually generate CI for any confidence level you want – these are just the most common

40. 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? • Increase sample size (This will decrease variability) • Decrease variability through more careful assessment • and measurement practices (minimize noise) . • Decrease level of confidence 95% 95%

41. ? ? Mean = 50Standard deviation = 10 Find the scores for the middle 95% 95% x = mean ± (z)(standard deviation) 30.4 69.6 .9500 Please note: We will be using this same logic for “confidence intervals” .4750 .4750 ? 1) Go to z table - find z score for for area .4750 z = 1.96 2) x = mean + (z)(standard deviation) x = 50 + (-1.96)(10) x = 30.4 30.4 3) x = mean + (z)(standard deviation) x = 50 + (1.96)(10) x = 69.6 69.6 Scores 30.4 - 69.6 capture the middle 95% of the curve

42. ? ? Mean = 50Standard deviation = 10 n = 100 s.e.m. = 1 Confidence intervals σ 95% standard error of the mean = Find the scores for the middle 95% n √ 48.04 51.96 For “confidence intervals” same logic – same z-score But - we’ll replace standard deviation with the standard error of the mean .9500 .4750 .4750 ? 10 = 100 √ x = mean ± (z)(s.e.m.) x = 50 + (1.96)(1) x = 51.96 x = 50 + (-1.96)(1) x = 48.04 95% Confidence Interval is captured by the scores 48.04 – 51.96

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

44. Thank you! See you next time!!