Describing the Relation between Two Variables

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1 Copyright 010 Pearso Educatio, Ic. Tables ad Formulas for Sulliva, Statistics: Iformed Decisios Usig Data 010 Pearso Educatio, Ic Chapter Orgaizig ad Summarizig Data Relative frequecy = frequecy sum of all frequecies Class midpoit: The sum of cosecutive lower class limits divided by. Chapter 3 Numerically Summarizig Data Populatio Mea: m = gx i N Weighted Mea: x w = gw ix i gw i Sample Mea: x = gx i Rage = Largest Data Value - Smallest Data Value Populatio Variace: s = g1x i - m gx i - 1 gx i N = N N Sample Variace: s = g1x i - x gx i - 1gx i = Populatio Stadard Deviatio: s = s Sample Stadard Deviatio: s = s Empirical Rule: If the shape of the distributio is bellshaped, the Approximately 68% of the data lie withi 1 stadard deviatio of the mea Approximately 95% of the data lie withi stadard deviatios of the mea Approximately 99.7% of the data lie withi 3 stadard deviatios of the mea Populatio Mea from Grouped Data: m = gx if i gf i Sample Mea from Grouped Data: x = gx if i gf i Populatio Variace from Grouped Data: s = g1x i - m f i gf i = Sample Variace from Grouped Data: s = g1x i - m f i A gf i B - 1 Populatio z-score: Sample z-score: = z = x - x s Iterquartile Rage: IQR = Q 3 - Q 1 Lower fece = Q Lower ad Upper Feces: IQR Upper fece = Q IQR Five-Number Summary gx i f i - 1gx if i gx i f i - 1gx if i gf i gf i - 1 z = x - m s gf i gf i Miimum, Q 1, M, Q 3, Maximum CHAPTER 4 Describig the Relatio betwee Two Variables Correlatio Coefficiet: r = a a x i - x s x - 1 ba y i - y b s y Residual = observed y - predicted y = y - yn R = r for the least-squares regressio model yn = b 1 x + b 0 The equatio of the least-squares regressio lie is yn = b 1 x + b 0, where yn is the predicted value, is the slope, ad b 0 = y - b 1 x is the itercept. b 1 = r # s y s x The coefficiet of determiatio, R, measures the proportio of total variatio i the respose variable that is explaied by the least-squares regressio lie. CHAPTER 5 Probability Empirical Probability Classical Probability frequecy of E P1E L umber of trials of experimet umber of ways that E ca occur P1E = umber of possible outcomes = N1E N1S Additio Rule for Disjoit Evets P1E or F = P1E + P1F Additio Rule for Disjoit Evets P1E or F or G or Á = P1E + P1F + P1G + Á Geeral Additio Rule P1E or F = P1E + P1F - P1E ad F

2 Copyright 010 Pearso Educatio, Ic. Tables ad Formulas for Sulliva, Statistics: Iformed Decisios Usig Data 010 Pearso Educatio, Ic Complemet Rule P1E c = 1 - P1E Multiplicatio Rule for Idepedet Evets P1E ad F = P1E # P1F Multiplicatio Rule for Idepedet Evets P1E ad F ad G Á = P1E # P1F # P1G # Á Coditioal Probability Rule P1E ad F N1E ad F P1Fƒ E = = P1E N1E Geeral Multiplicatio Rule P1E ad F = P1E # P1Fƒ E Factorial! = # 1-1 # 1 - # Á # 3 # # 1 Permutatio of objects take r at a time: Combiatio of objects take r at a time:! C r = r!1 - r! Permutatios with Repetitio:! 1! #! # Á # k! P r =! 1 - r! CHAPTER 6 Discrete Probability Distributios Mea (Expected Value) of a Discrete Radom Variable m X = gx # P1x Variace of a Discrete Radom Variable s X = g1x - m # P1x = gx P1x - m X Biomial Probability Distributio Fuctio CHAPTER 7 P1x = C x p x 11 - p - x The Normal Distributio Stadardizig a Normal Radom Variable z = x - m s CHAPTER 8 Samplig Distributios Mea ad Stadard Deviatio of the Samplig Distributio of x m x = m ad s x = s Sample Proportio: pn = x Mea ad Stadard Deviatio of a Biomial Radom Variable m X = p s X = 4 p11 - p Poisso Probability Distributio Fuctio P1x = 1ltx e -lt x = 0, 1,, Á x! Mea ad Stadard Deviatio of a Poisso Radom Variable m X = lt s X = lt Fidig the Score: x = m + zs Mea ad Stadard Deviatio of the Samplig Distributio of pn m pn = p ad s pn = C p11 - p CHAPTER 9 Estimatig the Value of a Parameter Usig Cofidece Itervals Cofidece Itervals Sample Size A 11 - a # 100% cofidece iterval about m with s To estimate the populatio mea with a margi of error E kow is x ; z a/ # s. 1 at a level of cofidece: = a z # a/ s 11 - a # 100% E A 11 - a # b 100% cofidece iterval about m with s rouded up to the ext iteger. ukow is x ; t a/ # s. Note: t a/ is computed usig 1 To estimate the populatio proportio with a margi - 1 degrees of freedom. of error E at a 11 - a # 100% level of cofidece: A 11 - a # 100% cofidece iterval about p is = pn11 - pna z a/ rouded up to the ext iteger, E b pn11 - p p ; z a/ # where pn is a prior estimate of the populatio proportio,. C or = 0.5 a z a/ rouded up to the ext iteger whe A 11 - a # 100% cofidece iterval about s E b is o prior estimate of p is available. 1-1s 1-6 s 1s 6. x a/ x 1 - a/

3 Copyright 010 Pearso Educatio, Ic. Tables ad Formulas for Sulliva, Statistics: Iformed Decisios Usig Data 010 Pearso Educatio, Ic CHAPTER 10 Testig Claims Regardig a Parameter Test Statistics z 0 = x - m 0 s sigle mea, s kow 1 t 0 = x - m 0 s sigle mea, s ukow 1 z 0 = x 0 = pn - p 0 p p 0 C 1-1s s 0 CHAPTER 11 Ifereces o Two Samples Test Statistic for Matched-Pairs data where d is the mea ad s d is the stadard deviatio of the differeced data. Cofidece Iterval for Matched-Pairs data: s d ; t a/ # d 1 Note: is foud usig - 1 degrees of freedom. t a/ t 0 = d - m d s d1 Test Statistic Comparig Two Meas (Idepedet Samplig): t 0 = 1x 1 - x - 1m 1 - m + s C 1 Cofidece Iterval for the Differece of Two Meas (Idepedet Samples): s 1 s 1 1x 1 - x ; t a/ + s C 1 Note: t a/ is foud usig the smaller of 1-1 or - 1 degrees of freedom. Test Statistic Comparig Two Populatio Proportios z 0 = pn 1 - pn - (p 1 - p ) 4 pn11 - pn B 1 where Cofidece Iterval for the Differece of Two Proportios 1pN 1 - pn ; z a/ C Test Statistic for Comparig Two Populatio Stadard Deviatios F 0 = s 1 Fidig a Critical F for the Left Tail pn pn pn 11 - pn F 1 - a,1-1, - 1 = s 1 F a, - 1, 1-1 pn = x 1 + x 1 +. CHAPTER 1 Iferece o Categorical Data Expected Couts (whe testig for goodess of fit) E i = m i = p i for i = 1,, Á, k Expected Frequecies (whe testig for idepedece or homogeeity of proportios) 1row total1colum total Expected frequecy = table total Chi-Square Test Statistic x 0 = a 1observed - expected expected i = 1,, Á, k = a 1O i - E i All E i Ú 1 ad o more tha 0% less tha 5. E i CHAPTER 13 Comparig Three or More Meas Test Statistic for Oe-Way ANOVA Mea square due to treatmet F = Mea square due to error where = MST MSE Test Statistic for Tukey s Test after Oe-Way ANOVA q = 1x - x 1-1m - m 1 s # a = b A 1 x - x 1 s # a b A 1 MST = 11x 1 - x + 1x - x + Á + k 1x k - x k - 1 MSE = 1 1-1s s + Á + 1 k - 1s k - k

4 Copyright 010 Pearso Educatio, Ic. Tables ad Formulas for Sulliva, Statistics: Iformed Decisios Usig Data 010 Pearso Educatio, Ic CHAPTER 14 Iferece o the Least-squares Regressio Model ad Multiple Regressio Stadard Error of the Estimate Stadard error of s e = C g1y i - yn i - Test statistic for the Slope of the Least-Squares Regressio Lie t 0 = b 1 s b1 = 4 g1x i - x b 1 - b 1 s e 4 g1x i - x = b 1 - b 1 s b1 Cofidece Iterval for the Slope of the Regressio Lie b 1 ; t a/ # = C g residuals - where t a/ is computed with - degrees of freedom. s e s e 4 g1x i - x Cofidece Iterval about the Mea Respose of y, yn yn ; t a/ # sec 1 + 1x - x g1x i - x where x is the give value of the explaatory variable ad t a/ is the critical value with - degrees of freedom. Predictio Iterval about a Idividual Respose, yn yn ; t a/ # se C x - x g1x i - x where x is the give value of the explaatory variable ad t a/ is the critical value with - degrees of freedom. CHAPTER 15 Noparametric Statistics Test Statistic for a Rus Test for Radomess Small-Sample Case If 1 0 ad 0, the test statistic i the rus test for radomess is r, the umber of rus. Large-Sample Case If or 7 0, the test statistic is z 0 = r - m r s r where m r = ad s r = B Test Statistic for a Oe-Sample Sig Test Small-Sample Case ( 5) Two-Tailed Left-Tailed Right-Tailed H o : M = M o H o : M = M o H 1 : M Z M o H 1 : M 6 M o H 1 : M 7 M o The test statistic, k, will The test statistic, The test statistic, be the smaller of the k, will be the k, will be the umber of mius sigs umber of umber of or plus sigs. plus sigs. mius sigs. Large-Sample Case ( >5) The test statistic,, is 1k z 0 = 1 H o : M = M o where is the umber of mius ad plus sigs ad k is obtaied as described i the small sample case. Test Statistic for the Wilcoxo Matched-Pairs Siged-Raks Test Small-Sample Case ( 30) Two-Tailed Left-Tailed Right-Tailed H o : M D = 0 H 1 : M D Z 0 H o : M D = 0 H 1 : M D 6 0 z 0 H o : M D = 0 H o : M D 7 0 ƒ ƒ Test Statistic: T is the Test Statistic: Test Statistic: smaller of T + or T - T = T + T = T - Large-Sample Case (>30) T - 4 z 0 = C 4 where T is the test statistic from the small-sample case. Test Statistic for the Ma Whitey Test Small-Sample Case ( 1 0 ad 0) If S is the sum of the raks correspodig to the sample from populatio X, the the test statistic, T, is give by T = S Note: The value of S is always obtaied by summig the raks of the sample data that correspod to M X i the hypothesis. Large-Sample Case ( 1 >0) or ( >0) T - 1 z 0 = B 1 Test Statistic for Spearma s Rak Correlatio Test 6gd i r s = where d i = the differece i the raks of the two observatios i the ordered pair. Test Statistic for the Kruskal Wallis Test where i th H = R i 1 1 N1N + 1 a 1 = N1N + 1 B R 1 + R 1 BR i - i1n + 1 i is the sum of the raks i the ith sample. R + Á + R k R - 31N + 1 k

5 Copyright 010 Pearso Educatio, Ic. Tables ad Formulas for Sulliva, Statistics: Iformed Decisios Usig Data 010 Pearso Educatio, Ic Table I Radom Numbers Colum Number Row Number Table II Critical Values for Correlatio Coefficiet

6 Copyright 010 Pearso Educatio, Ic. Tables ad Formulas for Sulliva, Statistics: Iformed Decisios Usig Data 010 Pearso Educatio, Ic Area z Table V Stadard Normal Distributio z Cofidece Iterval Critical Values, z A/ Level of Cofidece Critical Value, z A/ 0.90 or 90% or 95% or 98% or 99%.575 Hypothesis Testig Critical Values Level of Sigificace, A Left Tailed Right Tailed Two-Tailed

7 Copyright 010 Pearso Educatio, Ic. Tables ad Formulas for Sulliva, Statistics: Iformed Decisios Usig Data 010 Pearso Educatio, Ic Area i right tail t Table VI t-distributio Area i Right Tail df z

8 Copyright 010 Pearso Educatio, Ic. Tables ad Formulas for Sulliva, Statistics: Iformed Decisios Usig Data 010 Pearso Educatio, Ic Degrees of Freedom Table VII Chi-Square (X ) Distributio Area to the Right of Critical Value Right tail Left tail Area 1 a Two tails X a a The area to the right of this value is a. X 1 a a The area to the right of this value is 1 a. X 1 a a X a a The area to the right of this value is a. The area to the right a of this value is 1.

9 Area F Table VIII F-Distributio Critical Values Degrees of Freedom i the Numerator Area i Right Tail Degrees of Freedom i the Deomiator Copyright (c) 010 Pearso Educatio, Ic A 15

10 Area F Table VIII (cotiued) F-Distributio Critical Values Degrees of Freedom i the Numerator Area i Right Tail Degrees of Freedom i the Deomiator A 16 Copyright (c) 010 Pearso Educatio, Ic

11 Table VIII (cotiued) F-Distributio Critical Values Degrees of Freedom i the Numerator Area i Right Tail Degrees of Freedom i the Deomiator Copyright (c) 010 Pearso Educatio, Ic A 17

12 Table VIII (cotiued) F-Distributio Critical Values Degrees of Freedom i the Numerator Area i Right Tail Degrees of Freedom i the Deomiator A 18 Copyright (c) 010 Pearso Educatio, Ic

Tables and Formulas for Sullivan, Fundamentals of Statistics, 2e Pearson Education, Inc.

Tables and Formulas for Sullivan, Fundamentals of Statistics, 2e Pearson Education, Inc. Table ad Formula for Sulliva, Fudametal of Statitic, e. 008 Pearo Educatio, Ic. CHAPTER Orgaizig ad Summarizig Data Relative frequecy frequecy um of all frequecie Cla midpoit: The um of coecutive lower