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

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1 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 cla limit divided by. CHAPTER 3 umerically Summarizig Data Populatio Mea: m gx i Weighted Mea: x w gw ix i gw i Sample Mea: Rage Larget Data Value - Smallet Data Value Populatio Variace: g1x i - m gx i - 1 gx i Sample Variace: x gx i g1x i - x - 1 Populatio Stadard Deviatio Sample Stadard Deviatio: Empirical Rule: If the hape of the ditributio i bellhaped, the Approximately 68% of the data lie withi 1 tadard deviatio of the mea Approximately 95% of the data lie withi tadard deviatio of the mea Approximately 99.7% of the data lie withi 3 tadard deviatio of the mea Chebyhev Iequality: For ay data et, regardle of the hape of the ditributio, at leat a1-1 of k b100% the obervatio will lie withi k tadard deviatio of the mea where k i ay umber greater tha 1. Populatio Mea from Grouped Data: m gx if i Sample Mea from Grouped Data: x gx if i gx i - 1gx i - 1 Populatio Variace from Grouped Data g1x i - m f i Sample Variace from Grouped Data: g1x i - m f i A B - 1 Populatio Z-core: Sample Z-core: Percetile of x z x - x Determiig the kth percetile: i a k b + 1. If i i 100 ot a iteger, fid the mea of the obervatio o either ide of i. Iterquartile Rage: IQR Q 3 - Q 1 Lower fece Q Lower ad Upper Fece: IQR Upper fece Q IQR Five-umber Summary gx i f i - 1gx if i gx i f i - 1gx if i - 1 z x - m umber of data value le tha x # 100 Miimum, Q 1, M, Q 3, Maximum

2 Table ad Formula for Sulliva, Fudametal of Statitic, e. 008 Pearo Educatio, Ic. CHAPTER 4 Decribig the Relatio betwee Two Variable Correlatio Coefficiet: r a a x i - x x - 1 ba y i - y b y The equatio of the leat-quare regreio lie i where i the predicted value, b 1 r # y y b 1 x + b 0, y x i the lope, ad b 0 y - b 1 x i the itercept. Reidual oberved y - predicted y y - y Coefficiet of Determiatio: R the percet of total variatio i the repoe variable that i explaied by the leat-quare regreio lie. R r for the leat-quare regreio model y b 1 x + b 0 CHAPTER 5 Probability Empirical Probability Claical Probability umber of way that E ca occur P1E umber of poible outcome Additio Rule for Dijoit Evet Additio Rule for Dijoit Evet P1E or F or G or Á P1E + P1F + P1G + Á Geeral Additio Rule P1E or F P1E + P1F - P1E ad F Complemet Rule frequecy of E P1E L umber of trial of experimet P1E or F P1E + P1F P1E c 1 - P1E Multiplicatio Rule for Idepedet Evet P1E ad F P1E # P1F 1E 1S Multiplicatio Rule for Idepedet Evet Coditioal Probability Rule Geeral Multiplicatio Rule Factorial P1E ad F ad G Á P1E # P1F # P1G # Á P1E ad F 1E ad F P1Fƒ E P1E 1E P1E ad F P1E # P1Fƒ E! # - 1 # - # ### # 3 # # 1 Permutatio of object take r at a time: Combiatio of object take r at a time:! C r r! - r! Permutatio with Repetitio: 1 of oe type, of a ecod type, with Á Á, + k! 1! #! ##### k! P r! - r!

3 Table ad Formula for Sulliva, Fudametal of Statitic, e. 008 Pearo Educatio, Ic. CHAPTER 6 Dicrete Probability Ditributio Mea (Expected Value) of a Dicrete Radom Variable m X gx # P1X x Variace of a Dicrete Radom Variable X g1x - m # P1x gx P1x - mx Mea of a Biomial Radom Variable m X p Stadard Deviatio of a Biomial Radom Variable X 4 p11 - p Biomial Probability Ditributio Fuctio P1x C x p x 11 - p - x CHAPTER 7 The ormal Ditributio Stadardizig a ormal Radom Variable Z X - m Fidig the Score: X m + Z CHAPTER 8 Samplig Ditributio Mea ad Stadard Deviatio of the Samplig Ditributio of x m x m ad x Sample Proportio: p x Mea ad Stadard Deviatio of the Samplig Ditributo of p m p p ad p C p11 - p Stadardizig a ormal Radom Variable Z x - m Z p - p p11 - p C CHAPTER 9 Etimatig the Value of a Parameter Uig Cofidece Iterval Cofidece Iterval Sample Size A 11 - a # 100% cofidece iterval about m with To etimate the populatio mea with a margi of error E kow i x ; z #, provided the populatio from at a 11 - a # 100% level of cofidece require a ample a z # a which the ample wa draw i ormal or the ample ize of ize a rouded up to the ext iteger. E b i large Ú 30. To etimate the populatio proportio with a margi of A 11 - a # 100% cofidece iterval about m with error E at a 11 - a # 100% level of cofidece require a ukow i x ; t # a provided the populatio from, z a ample of ize p11 - pa rouded up to the E b which the ample wa draw i ormal or the ample ize ext iteger, where p i a prior etimate of the populatio i large Ú 30. ote: t a i computed uig - 1 proportio. degree of freedom. A 11 - a # To etimate the populatio proportio with a margi of 100% cofidece iterval about p i error E at a 11 - a # 100% level of cofidece require a p11 - p z p ; z # a a, provided p11 - p Ú 10. ample of ize 0.5 a rouded up to the ext C E b iteger whe o prior etimate of p i available.

4 Table ad Formula for Sulliva, Fudametal of Statitic, e. 008 Pearo Educatio, Ic. CHAPTER 10 Hypothei Tet Regardig a Parameter Tet Statitic z 0 x - m 0, provided that the populatio from which the ample wa draw i ormal or the ample ize i large Ú 30. t 0 x - m 0 follow Studet t-ditributio with - 1 p - p 0 z 0, provided that p p 0 Ú 10 ad p p 0 C the ample ize i le tha 5% of the populatio ize degree of freedom, provided that the populatio from which the ample wa draw i ormal or the ample ize i large Ú 30. CHAPTER 11 Iferece o Two Sample Tet Statitic for Matched-Pair data where d i the mea ad d i the tadard deviatio of the differeced data. Cofidece Iterval for Matched-Pair data: ote: i foud uig - 1 degree of freedom. t a t 0 d d Lower boud: d - t a Upper boud: d + t a Tet Statitic Comparig Two Mea (Idepedet Samplig): t 0 1x 1 - x - 1m 1 - m 1 + C 1 # d # d Tet Statitic Comparig Two Populatio Proportio where p x 1 + x 1 +. p 1 - p z 0 4 p11 - p B 1 Cofidece Iterval for the Differece of Two Proportio Lower boud: 1p 1 - p - z a C p p p 11 - p Upper boud: 1p 1 - p + z a C p p p 11 - p Cofidece Iterval for the Differece of Two Mea (Idepedet Sample): 1 Lower boud: 1x 1 - x - t a + C 1 1 Upper boud: 1x 1 - x + t a + C 1 ote: t a i foud uig the maller of 1-1 or - 1 degree of freedom.

5 Table ad Formula for Sulliva, Fudametal of Statitic, e. 008 Pearo Educatio, Ic. CHAPTER 1 Additioal Iferetial Procedure Expected Cout (whe tetig for goode of fit) Expected Frequecie (whe tetig for idepedece or homogeeity of proportio) Chi-Square Tet Statitic (1) All expected frequecie are greater tha or equal to 1 ad () o more tha 0% of the expected frequecie are le tha 5. Ue k - 1 degree of freedom for goode of fit. Ue 1r - 11c - 1 degree of freedom whe tetig for idepedece or homogeeity of proportio (r i the umber of row, c i the umber of colum). Stadard Error of the Etimate Stadard error of E i m i p i for i 1,, Á, k 1row total1colum total Expected frequecy table total x a 1oberved - expected expected i 1,, Á, k e C g1y i - y i - Tet tatitic for the Slope of the Leat-Square Regreio Lie t 0 b 1 e b1 4 g1x i - x a 1O i - E i E i C g reidual - b 1 - b 1 e 4 g1x i - x b 1 - b 1 b1 Cofidece Iterval for the Slope of the Regreio Lie A 11 - a # 100% cofidece iterval for the lope of the true regreio lie, b 1, i give by: where i computed with - degree of freedom. t a Cofidece Iterval about the Mea Repoe of y, y A 11 - a # 100% cofidece iterval for the mea repoe of y, y, i give by Lower boud: y - t a Upper boud: y + t a where i the give value of the explaatory variable ad t a i the critical value with - degree of freedom. x Predictio Iterval about a Idividual Repoe, y A 11 - a # 100% predictio iterval for the idividual repoe of y, y, i give by Lower Boud: y - t # a e C x - x g1x i - x Upper Boud: y + t # a e C x - x g1x i - x where i the give value of the explaatory variable ad t a i the critical value with - degree of freedom. x Lower boud: b 1 - t # e a 4 g1x i - x Upper boud: b 1 + t # a e 4 g1x i - x # ec 1 + 1x - x g1x i - x # ec 1 + 1x - x g1x i - x

Describing the Relation between Two Variables

Describing the Relation between Two Variables 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