Pearson Edexcel Level 3 Advanced Subsidiary and Advanced GCE in Statistics

Transcription

1 Pearso Edecel Level 3 Advaced Subsidiary ad Advaced GCE i Statistics Statistical formulae ad tables For first certificatio from Jue 018 for: Advaced Subsidiary GCE i Statistics (8ST0) For first certificatio from Jue 019 for: Advaced GCE i Statistics (9ST0) This copy is the property of Pearso. It is ot to be removed from the eamiatio room or marked i ay way. P58394A 017 Pearso Educatio Ltd. 1/1/1/1/1/1/1/

2 Edecel, BTEC ad LCCI qualificatios Edecel, BTEC ad LCCI qualificatios are awarded by Pearso, the UK s largest awardig body offerig academic ad vocatioal qualificatios that are globally recogised ad bechmarked. For further iformatio, please visit our qualificatio website at qualificatios.pearso.com. Alteratively, you ca get i touch with us usig the details o our cotact us page at qualificatios.pearso.com/cotactus About Pearso Pearso is the world s leadig learig compay, with 35,000 employees i more tha 70 coutries workig to help people of all ages to make measurable progress i their lives through learig. We put the learer at the cetre of everythig we do, because wherever learig flourishes, so do people. Fid out more about how we ca help you ad your learers at qualificatios.pearso.com All iformatio i this documet is correct at time of publicatio. ISBN All the material i this publicatio is copyright Pearso Educatio Limited 017

3 Cotets 1 Itroductio 1 AS Level i Statistics 3 A Level i Statistics 4 4 Statistical Tables 8 Table 1: Cumulative Biomial Distributio Fuctio 8 Table : Cumulative Poisso Distributio Fuctio 15 Table 3: Normal Distributio Fuctio 17 Table 4: Percetage Poits of the Normal Distributio 18 Table 5: Percetage Poits of the Studet s t-distributio 19 Table 6: Percetage Poits of the χ Distributio 0 Table 7: Percetage Poits of the F-distributio 1 Table 8: Critical Values of the Product Momet Correlatio Coefficiet 3 Table 9: Critical Values of Spearma s Rak Correlatio Coefficiet 4 Table 10: Critical Values of the Wilcoo Siged-Rak Statistic 5 Table 11: Critical Values of the Wilcoo Rak-Sum 6

4

5 1 Itroductio The formulae i this booklet have bee arraged by qualificatio. Studets sittig AS Statistics papers should refer to Sectio, pages 3. Studets sittig A Level Statistics papers should refer to Sectio 3, pages 4 7. Pearso Edecel Level 3 Advaced Subsidiary ad Advaced GCE i Statistics Statistical Formulae ad Tables Issue 1 August 017 Pearso Educatio Limited 017 1

6 AS Level i Statistics Populatio variace, σ, = N 1 μ = μ N ( ) Populatio stadard deviatio, σ, = N 1 μ = μ N ( ) Sample variace = 1 1 ( ) = 1 1 ( ) Sample stadard deviatio = 1 1 ( ) = 1 1 ( ) Biomial probability calculatios: P X p ( = ) = 1 p ( ) Mea = p Variace = p(1 p) For a radom sample of observatios from N ( μ, σ ) X μ ~ N(0, 1) σ Test statistic for a biomial proportio usig ormal distributio: pˆ p p ( 1 p) ~ N(0, 1) Pearso Edecel Level 3 Advaced Subsidiary ad Advaced GCE i Statistics Statistical Formulae ad Tables Issue 1 August 017 Pearso Educatio Limited 017

7 Product momet correlatio coefficiet: r = S S y S yy = ( ) ( y y) i i { ( y y i ) }{ ( i ) } = i ( ) y i ( i ) y i i i yi ( ) ( y ) i Coefficiets for least squares regressio lie: least squares regressio lie of y o is y = a + b, where a = y b Sy the regressio coefficiet of y o is b = = S Test for associatio: ( O E i i) E i is approimately distributed as χ ( y y i )( i ) ( i ) Pearso Edecel Level 3 Advaced Subsidiary ad Advaced GCE i Statistics Statistical Formulae ad Tables Issue 1 August 017 Pearso Educatio Limited 017 3

8 3 A Level i Statistics Populatio variace, σ, = N 1 μ = μ N ( ) Populatio stadard deviatio, σ, = N 1 μ = μ N ( ) Sample variace, s, = 1 1 ( ) = 1 1 ( ) Sample stadard deviatio, s, = 1 1 ( ) = 1 1 ( ) Biomial probability calculatios: P X p ( = ) = 1 p ( ) Biomial mea = p Biomial variace = p (1 p) For a radom sample of observatios from N ( μ, σ ) X μ ~ N(0, 1) σ Test statistic for a biomial proportio usig ormal distributio: pˆ p p ( 1 p) ~ N(0, 1) 4 Pearso Edecel Level 3 Advaced Subsidiary ad Advaced GCE i Statistics Statistical Formulae ad Tables Issue 1 August 017 Pearso Educatio Limited 017

9 Product momet correlatio coefficiet: r = S S y S yy = ( ) ( y y) i i { ( y y i ) }{ ( i ) } = i ( ) y i ( i ) y i i i yi ( ) ( y ) i Coefficiets for least squares regressio lie: least squares regressio lie of y o is y = a + b, where a = y b Sy the regressio coefficiet of y o is b = = S Bayes theorem for up to three evets: P( A P B A j) ( j) P( A B j ) = P A P B A i= 1 ( i) ( i) ( y y i )( i ) ( i ) The Poisso probability formula: ( = ) = e P X λ λ! Poisso mea = λ Poisso variace = λ The epoetial cumulative probability formula: P(X ) = 1 e λ Epoetial mea = 1 λ 1 Epoetial variace = λ EaX ( ± by) = ae( X) ± be( Y ) ( ) = ( ) + ( ) Var ax ± by a Var X b Var Y, for idepedet variables X ad Y Pearso Edecel Level 3 Advaced Subsidiary ad Advaced GCE i Statistics Statistical Formulae ad Tables Issue 1 August 017 Pearso Educatio Limited 017 5

10 For a radom sample of observatios from N (μ, σ ) X μ S ~ t 1 (also valid i matched-pairs situatios) For a radom sample of observatios from N (μ, σ ) ad, idepedetly, a radom sample of y observatios from N (μ y, σ y ) ( X Y) ( μ μ y) σ σ + y y ~ N(0, 1) For a radom sample of observatios from N (μ, σ ) ad, idepedetly, a radom sample of y observatios from N (μ y, σ y ) where σ = σ y = σ (ukow) ( X Y) ( μ μ y) S p = 1 1 S p + ( ) + y 1 S 1 S + ~ t + y where y y y ( ) Test statistic for the differece i two biomial proportios: p p 1 stadard error where stadard error = p p 1 1 ( 1 ) + 1 where p = p + p Test for associatio ad goodess of fit test: ( O E i i) E i is approimately distributed as χ 6 Pearso Edecel Level 3 Advaced Subsidiary ad Advaced GCE i Statistics Statistical Formulae ad Tables Issue 1 August 017 Pearso Educatio Limited 017

11 Aalysis of variace (oe-way ad two-way): oe-factor model ij = μ + α i + ε ij, where ε ij ~ N (0, σ ) total sum of squares SS T ij i j = T betwee groups sum of squares SS B Ti T = i i two-factor model (with m rows ad colums) ij = μ + α i + β j + ε ij, where ε ij ~ N (0, σ ) total sum of squares SS T ij i j = T m betwee rows sum of squares SS R Ri T = m i betwee colums sum of squares SS C C j T = m m j Cohe s d formula: d = ( 1 ) s where s = ( ) + ( ) 1 s 1 s Pearso Edecel Level 3 Advaced Subsidiary ad Advaced GCE i Statistics Statistical Formulae ad Tables Issue 1 August 017 Pearso Educatio Limited 017 7

12 4 Statistical Tables Table 1: Cumulative Biomial Distributio Fuctio The tabulated value is P(X ), where X has a biomial distributio with parameters ad p. p p = = = = = = = Pearso Edecel Level 3 Advaced Subsidiary ad Advaced GCE i Statistics Statistical Formulae ad Tables Issue 1 August 017 Pearso Educatio Limited 017

13 p p = = = = Pearso Edecel Level 3 Advaced Subsidiary ad Advaced GCE i Statistics Statistical Formulae ad Tables Issue 1 August 017 Pearso Educatio Limited 017 9

14 p p = = = Pearso Edecel Level 3 Advaced Subsidiary ad Advaced GCE i Statistics Statistical Formulae ad Tables Issue 1 August 017 Pearso Educatio Limited 017

15 p p = = Pearso Edecel Level 3 Advaced Subsidiary ad Advaced GCE i Statistics Statistical Formulae ad Tables Issue 1 August 017 Pearso Educatio Limited

16 p p = Pearso Edecel Level 3 Advaced Subsidiary ad Advaced GCE i Statistics Statistical Formulae ad Tables Issue 1 August 017 Pearso Educatio Limited 017

17 p p = Pearso Edecel Level 3 Advaced Subsidiary ad Advaced GCE i Statistics Statistical Formulae ad Tables Issue 1 August 017 Pearso Educatio Limited

18 p p = Pearso Edecel Level 3 Advaced Subsidiary ad Advaced GCE i Statistics Statistical Formulae ad Tables Issue 1 August 017 Pearso Educatio Limited 017

19 Table : Cumulative Poisso Distributio Fuctio The tabulated value is P(X ), where X has a Poisso distributio with mea λ. λ λ λ λ λ λ Pearso Edecel Level 3 Advaced Subsidiary ad Advaced GCE i Statistics Statistical Formulae ad Tables Issue 1 August 017 Pearso Educatio Limited

20 λ λ Pearso Edecel Level 3 Advaced Subsidiary ad Advaced GCE i Statistics Statistical Formulae ad Tables Issue 1 August 017 Pearso Educatio Limited 017

21 Table 3: Normal Distributio Fuctio The table gives the probability, p, that a ormally distributed radom variable Z, with mea = 0 ad variace = 1, is less tha or equal to z. z z Pearso Edecel Level 3 Advaced Subsidiary ad Advaced GCE i Statistics Statistical Formulae ad Tables Issue 1 August 017 Pearso Educatio Limited

22 Table 4: Percetage Poits of the Normal Distributio The table gives the values of z satisfyig P(Z z) = p, where Z is the ormally distributed radom variable with mea = 0 ad variace = 1. p p p p Pearso Edecel Level 3 Advaced Subsidiary ad Advaced GCE i Statistics Statistical Formulae ad Tables Issue 1 August 017 Pearso Educatio Limited 017

23 Table 5: Percetage Poits of the Studet s t-distributio The table gives the values of satisfyig P(X ) = p, where X is a radom variable havig the Studet s t-distributio with ν degrees of freedom. p p ν ν Pearso Edecel Level 3 Advaced Subsidiary ad Advaced GCE i Statistics Statistical Formulae ad Tables Issue 1 August 017 Pearso Educatio Limited

24 Table 6: Percetage Poits of the χ Distributio The table gives the values of satisfyig P(X ) = p, where X is a radom variable havig the χ distributio with ν degrees of freedom. p p ν ν Pearso Edecel Level 3 Advaced Subsidiary ad Advaced GCE i Statistics Statistical Formulae ad Tables Issue 1 August 017 Pearso Educatio Limited 017

25 Table 7: Percetage Poits of the F-distributio The tables give the values of satisfyig P(X ) = p, where X is a radom variable havig the F-distributio with ν 1 degrees of freedom i the umerator ad ν degrees of freedom i the deomiator. F-distributio ( p = 0.995) Use for oe-tail tests at sigificace level 0.5% or two-tail tests at sigificace level 1%. ν ν 1 ν ν F-distributio ( p = 0.99) Use for oe-tail tests at sigificace level 1% or two-tail tests at sigificace level %. ν ν 1 ν ν Pearso Edecel Level 3 Advaced Subsidiary ad Advaced GCE i Statistics Statistical Formulae ad Tables Issue 1 August 017 Pearso Educatio Limited 017 1