EM365_7-2_MidtermReview

Midterm Review Paul T. Grogan, Ph.D. Assistant Professor School of Systems and Enterprises EM 365: Statistics for Engineering Management Copyright © 2022 Paul T. Grogan | Creative Commons BY-NC 4.0 License 1

Midterm Format ▪ 100-minute individual exam worth 100 points ▪ ~6 multi-part questions ▪ Bring any printed/written materials (notes, textbook, readings, homework, lecture notes, practice exams, etc.) ▪ Bring calculator (four-function or graphing) ▪ No phone/computer ▪ No statistical software ▪ Topics cover chapters 1-10: - Probability theory (counting and laws/theorems) - Random variables and distributions - Inference for one population - Population mean ( 𝑧 , ? ) - Population variance ( 𝜒 2 ) - Inference for two populations - Difference in population means ( 𝑧 , ? ) - Differences in related population means ( ? ) - Difference in population variances ( ? ) 2 Copyright © 2022 Paul T. Grogan | Creative Commons BY-NC 4.0 License

Reference Slides 4 Copyright © 2022 Paul T. Grogan | Creative Commons BY-NC 4.0 License

Descriptive Statistics Population Statistics ? 𝑥 = 1 ? ෍ 𝑖=1 𝑁 𝑥 𝑖 𝜎 𝑥 2 = 1 ? ෍ 𝑖=1 𝑁 𝑥 𝑖 − ? 𝑥 2 Sample Statistics ҧ𝑥 = 1 ? ෍ 𝑖=1 𝑁 𝑥 𝑖 ? 𝑥 2 = 1 ? − 1 ෍ 𝑖=1 𝑁 𝑥 𝑖 − ҧ𝑥 2 5 Copyright © 2022 Paul T. Grogan | Creative Commons BY-NC 4.0 License

Probability Laws ▪ General Law of Addition: 𝑃 ? ∪ ? = 𝑃 ? + 𝑃 ? − 𝑃 ? ∩ ? - ? and ? mutually exclusive: 𝑃 ? ∪ ? = 𝑃 ? + 𝑃 ? ▪ General Law of Multiplication: 𝑃 ? ∩ ? = 𝑃 ? ∙ 𝑃 ?|? = 𝑃 ? ∙ 𝑃 ?|? - ? and ? independent: 𝑃 ? ∩ ? = 𝑃 ? ∙ 𝑃 ? ▪ Law of Conditional Probability (+ Bayes Theorem): 𝑃 ? ? = 𝑃 ? ∩ ? 𝑃 ? = 𝑃 ? ? ⋅ 𝑃 ? 𝑃 ? ? 𝑃 ? + 𝑃 ? ¬? 𝑃 ¬? 7 Copyright © 2022 Paul T. Grogan | Creative Commons BY-NC 4.0 License

Counting ▪ ? × ? outcomes from combining two events with ? and ? outcomes, respectively ▪ ? 𝑘 outcomes from selecting 𝑘 items (with replacement) from ? alternatives ▪ ?! outcomes from arranging ? items ▪ 𝑁! 𝑁−𝑘 ! = 𝑁 𝑃 𝑘 outcomes from choosing and arranging a subset 𝑘 of ? items ▪ 𝑁! 𝑘! 𝑁−𝑘 ! = ? 𝑘 = 𝑁 𝐶 𝑘 outcomes from choosing a subset 𝑘 of ? items (order not important) 8 Copyright © 2022 Paul T. Grogan | Creative Commons BY-NC 4.0 License

Discrete Distributions 10 Name Parameters 𝒑 𝒙 𝝁 𝒙 = ? 𝑿 𝝈 𝒙 ? = Var 𝑿 Uniform Min ?, Max ? > ?, 1 ? − ? + 1 ? + ? 2 ? − ? + 1 2 − 1 12 Bernoulli Prob. 𝑝 > 0 ቊ 𝑝, 𝑥 = 1 1 − 𝑝, 𝑥 = 0 𝑝 𝑝 ⋅ 1 − 𝑝 Binomial Trials 𝑛 > 0, Prob. 𝑝 > 0 𝑛 𝑥 𝑝 𝑥 1 − 𝑝 𝑛−𝑥 𝑛 ∙ 𝑝 𝑛 ∙ 𝑝 ∙ (1 − 𝑝) Geometric Prob. 𝑝 > 0 𝑝 1 − 𝑝 𝑥−1 1 𝑝 1 − 𝑝 𝑝 2 Hyper- geometric Trials 𝑛 > 0, Pop. ? > 0, Succ. 0 < 𝐴 < ? 𝐴 𝑥 ∙ ? − 𝐴 𝑛 − 𝑥 ? 𝑛 𝑛 ∙ 𝐴 ? 𝑛 ∙ 𝐴 ? ∙ ? − 𝐴 ? ∙ ? − 𝑛 ? − 1 Poisson Rate ? > 0 ? 𝑥 ? −? 𝑥! ? ? Copyright © 2022 Paul T. Grogan | Creative Commons BY-NC 4.0 License

Continuous Distributions 11 Name Parameters 𝒇 𝒙 𝝁 𝒙 = ? 𝑿 𝝈 𝒙 ? = Var 𝑿 Uniform Min ?, Max ? > ? 1 ? − ? ? + ? 2 ? − ? 2 12 Normal Mean ?, Std. Dev. 𝜎 1 2𝜎 2 𝜋 ? − 𝑥−? 2 2𝜎 2 ? 𝜎 2 Standard Normal (none) 1 2𝜋 ? − 𝑥 2 2 0 1 Chi-squared Degrees of freedom 𝑘 ? 𝑥 = 1 2 𝑘 2 Γ 𝑘 2 𝑥 𝑘 2 −1 ? − 𝑥 2 𝑘 2𝑘 Exponential Rate ? ?? −?𝑥 1 ? 1 ? 2 Given any normally-distributed random variable ?~ normal (? 𝑥 , 𝜎 𝑥 ) , the Z-transformed random variable (below) follows a standard normal distribution ?~ normal (0,1) . ? = ? − ? 𝑥 𝜎 𝑥 Copyright © 2022 Paul T. Grogan | Creative Commons BY-NC 4.0 License

13 Test 𝑯 ? 𝑯 𝒂 Test Statistic Critical Value(s) 𝒑 -Value Two-tailed 𝒛 Test ? 𝑥 = ? 0 ? 𝑥 ≠ ? 0 𝑧 = ҧ𝑥 − ? 0 𝜎 𝑥 / ? 𝑧 𝛼/2 = ? 𝑧 −1 1 − 𝛼/2 𝑧 1−𝛼/2 = ? 𝑧 −1 𝛼/2 2 ⋅ 1 − ? z |𝑧| One-tailed 𝑧 Test (>) ? 𝑥 = ? 0 ? 𝑥 > ? 0 𝑧 = ҧ𝑥 − ? 0 𝜎 𝑥 / ? 𝑧 𝛼 = ? 𝑧 −1 1 − 𝛼 1 − ? z 𝑧 One-tailed 𝑧 Test (<) ? 𝑥 = ? 0 ? 𝑥 < ? 0 𝑧 = ҧ𝑥 − ? 0 𝜎 𝑥 / ? 𝑧 1−𝛼 = ? 𝑧 −1 𝛼 ? z 𝑧 Two-tailed ? Test ? 𝑥 = ? 0 ? 𝑥 ≠ ? 0 ? 𝑁−1 = ҧ𝑥 − ? 0 ? 𝑥 / ? ? 𝛼/2,𝑁−1 = ? 𝑡 𝑁−1 −1 1 − 𝛼/2 ? 1−𝛼/2,𝑁−1 = ? 𝑡 𝑁−1 −1 𝛼/2 2 ⋅ 1 − ? 𝑡 𝑁−1 |? 𝑁−1 | One-tailed ? Test (>) ? 𝑥 = ? 0 ? 𝑥 > ? 0 ? 𝑁−1 = ҧ𝑥 − ? 0 ? 𝑥 / ? ? 𝛼,𝑁−1 = ? 𝑡 𝑁−1 −1 1 − 𝛼 1 − ? 𝑡 𝑁−1 ? 𝑁−1 One-tailed ? Test (<) ? 𝑥 = ? 0 ? 𝑥 < ? 0 ? 𝑁−1 = ҧ𝑥 − ? 0 ? 𝑥 / ? ? 1−𝛼,𝑁−1 = ? 𝑡 𝑁−1 −1 𝛼 ? 𝑡 𝑁−1 ? 𝑁−1 One-tailed 𝝌 ? Test (>) 𝜎 𝑥 2 = 𝜎 0 2 𝜎 𝑥 2 > 𝜎 0 2 𝜒 𝑁−1 2 = ? − 1 ? 𝑥 2 𝜎 0 2 𝜒 𝛼,𝑁−1 2 = ? 𝜒 𝑁−1 2 −1 1 − 𝛼 1 − ? 𝜒 𝑁−1 2 𝜒 𝑁−1 2 Copyright © 2022 Paul T. Grogan | Creative Commons BY-NC 4.0 License

14 Test 𝑯 ? 𝑯 𝒂 Test Statistic 𝒑 -Value Two-tailed 𝑧 Test ? 1 − ? 2 = ? 0 ? 1 − ? 2 ≠ ? 0 𝑧 = ҧ𝑥 1 − ҧ𝑥 2 − ? 0 𝜎 1 2 /? 1 + 𝜎 2 2 /? 2 2 ⋅ 1 − ? 𝑧 𝑧 Two-tailed ? Test ( 𝜎 1 2 = 𝜎 2 2 ) ? 1 − ? 2 = ? 0 ? 1 − ? 2 ≠ ? 0 ? 𝑘 = ҧ𝑥 1 − ҧ𝑥 2 − ? 0 ? 1 2 ? 1 − 1 + ? 2 2 ? 2 − 1 ? 1 + ? 2 − 2 ⋅ 1 ? 1 + 1 ? 2 𝑘 = ? 1 + ? 2 − 2 2 ⋅ 1 − ? 𝑡 𝑘 |? 𝑘 | Two-tailed ? Test ( 𝜎 1 2 ≠ 𝜎 2 2 ) ? 1 − ? 2 = ? 0 ? 1 − ? 2 ≠ ? 0 ? 𝑘 = ҧ𝑥 1 − ҧ𝑥 2 − ? 0 ? 1 2 /? 1 + ? 2 2 /? 2 𝑘 = ? 1 2 /? 1 + ? 2 2 /? 2 2 ? 1 2 /? 1 2 ? 1 − 1 + ? 2 2 /? 2 2 ? 2 − 1 2 ⋅ 1 − ? 𝑡 𝑘 |? 𝑘 | Two-tailed ? Test (Paired) ? 1−2 = ? 0 ? 1−2 ≠ ? 0 ? 𝑘 = ҧ𝑥 1−2 − ? 0 ? 𝑥 1−2 / ? , 𝑘 = ? − 1 2 1 − ? 𝑡 𝑘 |? 𝑘 | One-tailed ? Test ( 𝜎 1 2 > 𝜎 2 2 ) 𝜎 1 2 = 𝜎 2 2 𝜎 1 2 > 𝜎 2 2 ? 𝑘 1 ,𝑘 2 = ? 1 2 /? 2 2 𝑘 1 = ? 1 − 1, 𝑘 2 = ? 2 − 1 1 − ? 𝐹 𝑘 1 ,𝑘 2 ? 𝑘 1 ,𝑘 2 Copyright © 2022 Paul T. Grogan | Creative Commons BY-NC 4.0 License

Areas of the Standard Normal Distribution 16 Table from Ken Black's Textbook

Student's t Distribution (and Standard Normal) 17 Table from Ken Black's Textbook

F Distribution 19 Table from Ken Black's Textbook

Sample Problems 20 Copyright © 2022 Paul T. Grogan | Creative Commons BY-NC 4.0 License