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QM 2113 -- Fall 2003 Statistics for Decision Making Probability Applications: The Normal Distribution Instructor: John Seydel, Ph.D. Student Objectives Discuss the characteristics of normally distributed random variables Calculate probabilities for normal random variables Apply normal distribution concepts to practical problems Administrative Items Exam1 rework: haven’t completed the grading Grading scheme (clarification/correction) 300 points: Exercises & quizzes 100 points: Midterm exam Actually 3 midterm exams Grade will be the best of the three Next one: October 20 (bivariate analysis, probability distributions, sampling concepts, intro to inference) 200 points: Final exam (comprehensive) Collect homework: exercises from text Now, a Review What is probability? What’s a probability distribution? What are some basic rules for working with probability? Basic notation Range of possible values Complement rule Additive rules All possible events More than one event What do we mean by a “normal distribution”? Questions About the Homework? Data analysis (Web Analytics case) Univariate Bivariate Quantitative variables Qualitative variables Normal distribution problems Basic calculations (1, 2) Applications (4, 5) Understanding the concepts (35, 36) If Something’s Normally Distributed It’s described by m (the population/process average) s (the population/process standard deviation) Histogram is symmetric Thus no skew (average = median) So P(x < m) = P(x > m) = . . . ? Shape of histogram can be described by f(x) = (1/s√2p)e-[(x-m)2/2s 2] We determine probabilities based upon distance from the mean (i.e., the number of standard deviations) A Sketch is Essential! Use to identify regions of concern Enables putting together results of calculations, lookups, etc. Doesn’t need to be perfect; just needs to indicate relative positioning Make it large enough to work with; needs annotation (probabilities, comments, etc.) Keep In Mind Probability = proportion of area under the normal curve What we get when we use tables is always the area between the mean and z standard deviations from the mean Because of symmetry P(x > m) = P(x < m) = 0.5000 Tables show probabilities rounded to 4 decimal places If z < -3.89 then probability ≈ 0.5000 If z > 3.89 then probability ≈ 0.5000 Theoretically, P(x = a) = 0 P(30 ≤ x ≤ 35) = P(30 < x < 35) Additional Exercises Refer to review handout for OM class Project planning Process capability analysis Inventory planning From text Airline flight management (#7) Theatre concessions planning (#13-15) Summary of Objectives Discuss the characteristics of normally distributed random variables Calculate probabilities for normal random variables Apply normal distribution concepts to practical problems