Cheat Sheet for Statistics Formulas and Calculator Instructions, Lecture notes of Technology

Download Cheat Sheet for Statistics Formulas and Calculator Instructions and more Lecture notes Technology in PDF only on Docsity! Page 1 of 2 I will provide these cheat sheets on the day of the final exam. Do not bring this copy to the final. Also, it is your responsibility to verify that the formulas are correct and notify me of any necessary corrections. Formulas Data Distribution ?̅?𝑥 = ∑𝑥𝑥 𝑛𝑛 sd = �∑(𝑥𝑥−?̅?𝑥)2 𝑛𝑛−1 Probability Distribution 𝜇𝜇 = ∑𝑥𝑥 ∙ 𝑝𝑝(𝑥𝑥) 𝜎𝜎 = �∑(𝑥𝑥 − 𝜇𝜇)2 ∙ 𝑝𝑝(𝑥𝑥) 𝑍𝑍 = 𝑥𝑥−𝜇𝜇 𝜎𝜎 , 𝑥𝑥 = 𝜇𝜇 + 𝑧𝑧 ∙ 𝜎𝜎 Empirical Rule = 68-95-99.7 Rule See also the common probability rules below. Regression Predicted 𝑦𝑦 = 𝑎𝑎+ 𝑏𝑏 ∙ 𝑥𝑥 𝑏𝑏 = 𝑟𝑟 ∙ 𝑠𝑠𝑦𝑦 𝑠𝑠𝑥𝑥 , 𝑎𝑎 = 𝑦𝑦� − 𝑏𝑏 ∙ ?̅?𝑥 Residual (predicted error) = 𝑦𝑦𝑜𝑜𝑜𝑜𝑠𝑠𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜 − 𝑦𝑦𝑝𝑝𝑜𝑜𝑜𝑜𝑜𝑜𝑝𝑝𝑝𝑝𝑝𝑝𝑜𝑜𝑜𝑜 S𝑜𝑜 = � SSE 𝑛𝑛−2 Distribution of Sample Proportions Mean = P, SE = �𝑃𝑃∙(1−𝑃𝑃) 𝑛𝑛 , 𝑧𝑧 = 𝑝𝑝�−𝑝𝑝 𝑆𝑆𝑆𝑆 Normal if 𝑛𝑛𝑝𝑝 ≥ 10 and (1 − 𝑝𝑝) ∙ 𝑛𝑛 ≥ 10 CI = 𝑝𝑝� ± 𝑍𝑍𝑐𝑐 ∙ SE Distribution of Sample Means Mean = 𝜇𝜇, 𝑧𝑧 = ?̅?𝑥−𝜇𝜇 𝑆𝑆𝑆𝑆 𝝈𝝈 known & population variable is normally distributed OR n > 30: Use Z-test SE = 𝜎𝜎 √𝑛𝑛 , CI = ?̅?𝑥 ± 𝑍𝑍𝑝𝑝 ∙ SE 𝝈𝝈 not known & population variable is normally distributed OR n > 30: Use T-test One sample: SE = 𝑠𝑠 √𝑛𝑛 , 𝑑𝑑𝑑𝑑 = 𝑛𝑛 − 1 , CI = ?̅?𝑥 ± 𝑇𝑇𝑝𝑝 ∙ 𝑆𝑆𝑆𝑆 Two independent samples: SE = �𝑠𝑠12 𝑛𝑛1 + 𝑠𝑠22 𝑛𝑛2 , 𝑑𝑑𝑑𝑑 (use technology) CI = (?̅?𝑥1 − ?̅?𝑥2) ± 𝑇𝑇𝑝𝑝 ∙ 𝑆𝑆𝑆𝑆 Chi-Square Tests All chi-square curves are skewed right; mean = 𝑑𝑑𝑑𝑑. Test statistic for all Chi-Square tests: 𝜒𝜒2 = (𝑜𝑜𝑜𝑜𝑠𝑠𝑜𝑜𝑜𝑜�𝑜𝑜𝑜𝑜−𝑜𝑜𝑥𝑥𝑝𝑝𝑜𝑜𝑝𝑝𝑝𝑝𝑜𝑜𝑜𝑜)2 𝑜𝑜𝑥𝑥𝑝𝑝𝑜𝑜𝑝𝑝𝑝𝑝𝑜𝑜𝑜𝑜 One-way table: 𝑑𝑑𝑑𝑑 = (𝑟𝑟 − 1) where r = number of categories. Two-way table: 𝑑𝑑𝑑𝑑 = (𝑟𝑟 − 1)(𝑐𝑐 − 1) where r = number of categories for one variable and c = number of categories for the other variable. Hypothesis Testing 𝐻𝐻0 : ≤,≥, = (then change to =) 𝐻𝐻𝑎𝑎: <, >,≠ (< or > is a one-tailed test; ≠ is a two-tailed test) Compare P-value to 𝛼𝛼 One-tailed test: 𝛼𝛼 = significance level Two-tailed test: 𝛼𝛼 = significance level 2 Z-test: SE = 𝜎𝜎 √𝑛𝑛 , 𝑧𝑧 = ?̅?𝑥−𝜇𝜇0 𝑆𝑆𝑆𝑆 1-sample T-test: SE = 𝑠𝑠 √𝑛𝑛 , 𝑇𝑇 = ?̅?𝑥−𝜇𝜇0 𝑆𝑆𝑆𝑆 2-sample T-test: SE = �𝑠𝑠1 2 𝑛𝑛1 + 𝑠𝑠2 2 𝑛𝑛2 𝑇𝑇 = (?̅?𝑥1−?̅?𝑥2)−(𝜇𝜇1−𝜇𝜇2) 𝑆𝑆𝑆𝑆 Paired T-test: SE = 𝑠𝑠 √𝑛𝑛 , 𝑇𝑇 = ?̅?𝑥−0 𝑆𝑆𝑆𝑆 Probability Rules a) For any event A, 0 ≤ 𝑃𝑃(𝐴𝐴) ≤ 1 . b) If S is a sample space, then P(S) = 1. c) The sum of the probabilities of all possible disjoint events in a sample space is 1. d) If A and B are disjoint events (no outcomes in common), then P(A or B) = P(A) + P(B). e) If A and B are NOT disjoint events (share at least one outcome), then P(A or B) = P(A) + P(B) – P( A and B). f) For any event A, P(not A) = 1 – P(A). g) If P(B|A) ≈ P(B) , then A and B are independent events. h) P(A and B) = P(A)∙P(B|A). i) When A and B are independent events then P(A and B) = P(A)∙P(B)