Pearson Edexcel Level 3 Advanced Subsidiary and Advanced ...

1 Pearson Edexcel Level 3 Advanced Subsidiary and Advanced GCE in StatisticsStatistical formulae and tables For first certification from June 2018 for: Advanced Subsidiary GCE in Statistics (8ST0)For first certification from June 2019 for: Advanced GCE in Statistics (9ST0)This copy is the property of Pearson . It is not to be removed from the examination room or marked in any 2017 Pearson Education , BTEC and LCCI qualificationsEdexcel, BTEC and LCCI qualifications are awarded by Pearson , the UK s largest awarding body offering academic and vocational qualifications that are globally recognised and benchmarked. For further information, please visit our qualification website at Alternatively, you can get in touch with us using the details on our contact us page at PearsonPearson is the world s leading learning company, with 35,000 employees in more than 70 countries working to help people of all ages to make measurable progress in their lives through learning .

2 We put the learner at the centre of everything we do, because wherever learning flourishes, so do people. Find out more about how we can help you and your learners at information in this document is correct at time of 978 1 4469 4764 7 All the material in this publication is copyright Pearson Education Limited 2017 Contents1 Introduction 12 AS Level in Statistics 23 A Level in Statistics 44 Statistical Tables 8 Table 1: Cumulative Binomial Distribution Function 8 Table 2: Cumulative Poisson Distribution Function 15 Table 3: Normal Distribution Function 17 Table 4: Percentage Points of the Normal Distribution 18 Table 5: Percentage Points of the Student s t-distribution 19 Table 6: Percentage Points of the 2 Distribution 20 Table 7.

3 Percentage Points of the F-distribution 21 Table 8: Critical Values of the Product Moment Correlation Coefficient 23 Table 9: Critical Values of Spearman s Rank Correlation Coefficient 24 Table 10: Critical Values of the Wilcoxon Signed-Rank Statistic 25 Table 11: Critical Values of the Wilcoxon Rank-Sum 26 Pearson Edexcel Level 3 Advanced Subsidiary and Advanced GCE in StatisticsStatistical Formulae and Tables Issue 1 August 2017 Pearson Education Limited 201711 IntroductionThe formulae in this booklet have been arranged by qualification. Students sitting AS Statistics papers should refer to Section 2, pages 2 3. Students sitting A Level Statistics papers should refer to Section 3, pages 4 Edexcel Level 3 Advanced Subsidiary and Advanced GCE in StatisticsStatistical Formulae and Tables Issue 1 August 2017 Pearson Education Limited 20172 AS Level in StatisticsPopulation variance, 2, =()2221x x NN = Population standard deviation, , =()2221x x NN = Sample variance =1111222nxxnnxx () = () Sample standard deviation =1111222nxxnnxx () = () Binomial probability calculations:PXxnxppxnx=()= () 1 Mean = npVariance = np(1 p)For a random sample of nx observations from ()2N, ()N0,1X n ~ N(0, 1)Test statistic for a binomial proportion using normal distribution.

4 ()() N0,11ppppn ~ N(0, 1) Pearson Edexcel Level 3 Advanced Subsidiary and Advanced GCE in StatisticsStatistical Formulae and Tables Issue 1 August 2017 Pearson Education Limited 20173 Product moment correlation coefficient:rSSSxxyyxxyyxyxyxyxxyyiiiiii i= = () () (){} (){}= () 22iiiiiinxxnyyn () () () 2222 Coefficients for least squares regression line: least squares regression line of y on x is y = a + bx , where a = y bxthe regression coefficient of y on x is bSSxxyyxxxyxxiii== () () () 2 Test for association:OEEiii () 2 is approximately distributed as 24 Pearson Edexcel Level 3 Advanced Subsidiary and Advanced GCE in StatisticsStatistical Formulae and Tables Issue 1 August 2017 Pearson Education Limited 20173 A Level in StatisticsPopulation variance, 2, =()2221x x NN = Population standard deviation, , =()2221x x NN = Sample variance, s2, =1111222nxxnnxx () = () Sample standard deviation, s, =1111222nxxnnxx () = () Binomial probability calculations:PXxnxppxnx=()= () 1 Binomial mean = npBinomial variance = np (1 p)For a random sample of nx observations from ()2N, ()N0,1X n ~ N(0, 1)Test statistic for a binomial proportion using normal distribution.

5 ()() N0,11ppppn ~ N(0, 1) Pearson Edexcel Level 3 Advanced Subsidiary and Advanced GCE in StatisticsStatistical Formulae and Tables Issue 1 August 2017 Pearson Education Limited 20175 Product moment correlation coefficient:rSSSxxyyxxyyxyxyxyxxyyiiiiii i= = () () (){} (){}= () 22iiiiiinxxnyyn () () () 2222 Coefficients for least squares regression line: least squares regression line of y on x is y = a + bx , where a = y bxthe regression coefficient of y on x is bSSxxyyxxxyxxiii== () () () 2 Bayes theorem for up to three events:PPPPPABABAABA jjjiiin()=() ()() ()= 1 The Poisson probability formula:()e!x PXxx ==Poisson mean = Poisson variance = The exponential cumulative probability formula:P(X x) = 1 e xExponential mean = 1 Exponential variance = 21 EaXbYaEXbEY ()=() ()VarVarVaraXbYaXbY ()=()+()22, for independent variables X and Y6 Pearson Edexcel Level 3 Advanced Subsidiary and Advanced GCE in StatisticsStatistical Formulae and Tables Issue 1 August 2017 Pearson Education Limited 2017 For a random sample of nx observations from N ( , 2)1nX tSn ~ tn 1 (also valid in matched-pairs situations)For a random sample of nx observations from N ( x, x2) and, independently, a random sample of ny observations from N ( y , y2)()()()22N0,1xyyxxyXY nn + ~ N(0, 1)For a random sample of nx observations from N ( x, x2) and, independently, a random sample of ny observations from N ( y , y2)

6 Where x2 = y2 = 2 (unknown)()()2211xynnxypxyXY tSnn+ + ~ tnx+ ny 2 whereSnSnSnnpxxyyxy222112= ()+ ()+ Test statistic for the difference in two binomial proportions:pp12 standard error where standard error = ppnn () + 11112where ppnpnnn= + +112212 Test for association and goodness of fit test:OEEiii () 2 is approximately distributed as 2 Pearson Edexcel Level 3 Advanced Subsidiary and Advanced GCE in StatisticsStatistical Formulae and Tables Issue 1 August 2017 Pearson Education Limited 20177 Analysis of variance (one-way and two-way):one-factor model xij = + i + ij , where ij ~ N (0, 2)total sum of squares SSxTnTijji= 22between groups sum of squares SSTnTnBiii= 22two-factor model (with m rows and n columns) xij = + i + j + ij , where ij ~ N (0, 2)total sum of squares SSxTmnTijji= 22between rows sum of squares SSRnTmnRii= 22between columns sum of squares SSCmTmnCjj= 22 Cohen s d formula:dxxs= ()12where snsnsnn= ()+ ()+ 112222121128 Pearson Edexcel Level 3 Advanced Subsidiary and Advanced GCE in StatisticsStatistical Formulae and Tables Issue 1 August 2017 Pearson Education Limited 20174 Statistical TablesTable 1.

7 Cumulative Binomial Distribution FunctionThe tabulated value is P(X x), where X has a binomial distribution with parameters n and = = = = = = = Edexcel Level 3 Advanced Subsidiary and Advanced GCE in StatisticsStatistical Formulae and Tables Issue 1 August 2017 Pearson Education Limited = = = = Edexcel Level 3 Advanced Subsidiary and Advanced GCE in StatisticsStatistical Formulae and Tables Issue 1 August 2017 Pearson Education Limited = = = Edexcel Level 3 Advanced Subsidiary and Advanced GCE in StatisticsStatistical Formulae and Tables Issue 1 August 2017 Pearson Education Limited = = Edexcel Level 3 Advanced Subsidiary and Advanced GCE in StatisticsStatistical Formulae and Tables Issue 1 August 2017 Pearson Education

8 Limited = Edexcel Level 3 Advanced Subsidiary and Advanced GCE in StatisticsStatistical Formulae and Tables Issue 1 August 2017 Pearson Education Limited = Edexcel Level 3 Advanced Subsidiary and Advanced GCE in StatisticsStatistical Formulae and Tables Issue 1 August 2017 Pearson Education Limited = Edexcel Level 3 Advanced Subsidiary and Advanced GCE in StatisticsStatistical Formulae and Tables Issue 1 August 2017 Pearson Education Limited 201715 Table 2: Cumulative Poisson Distribution FunctionThe tabulated value is P(X x), where X has a Poisson distribution with mean . Edexcel Level 3 Advanced Subsidiary and Advanced GCE in StatisticsStatistical Formulae and Tables Issue 1 August 2017 Pearson Education Limited 2017 Edexcel Level 3 Advanced Subsidiary and Advanced GCE in StatisticsStatistical Formulae and Tables Issue 1 August 2017 Pearson Education Limited 201717 Table 3: Normal Distribution FunctionThe table gives the probability, p, that a normally distributed random variable Z, with mean = 0 and variance = 1, is less than or equal to Edexcel Level 3 Advanced Subsidiary and Advanced GCE in StatisticsStatistical Formulae and Tables Issue 1 August 2017 Pearson Education Limited 2017 Table 4.

9 Percentage Points of the Normal DistributionThe table gives the values of z satisfying P(Z z) = p, where Z is the normally distributed random variable with mean = 0 and variance = Edexcel Level 3 Advanced Subsidiary and Advanced GCE in StatisticsStatistical Formulae and Tables Issue 1 August 2017 Pearson Education Limited 201719 Table 5: Percentage Points of the Student s t-distributionThe table gives the values of x satisfying P(X x) = p, where X is a random variable having the Student s t-distribution with degrees of Edexcel Level 3 Advanced Subsidiary and Advanced GCE in StatisticsStatistical Formulae and Tables Issue 1 August 2017 Pearson Education Limited 2017 Table 6: Percentage Points of the 2 DistributionThe table gives the values of x satisfying P(X x) = p, where X is a random variable having the 2 distribution with degrees of Edexcel Level 3 Advanced Subsidiary and Advanced GCE in StatisticsStatistical Formulae and Tables Issue 1 August 2017 Pearson Education Limited 201721 Table 7: Percentage Points of the F-distributionThe tables give the values of x satisfying P(X x) = p, where X is a random variable having the F-distribution with 1 degrees of freedom in the numerator and 2 degrees of freedom in the ( p = )Use for one-tail tests at significance Level or two-tail tests at significance Level 1%.

10 1123456789101112152025304050100 1 2 2116211 20000 21615 22500 23056 23437 23715 23925 24091 24224 24334 24426 24630 24836 24960 25044 25148 25211 25337 F-distribution ( p = )Use for one-tail tests at significance Level 1% or two-tail tests at significance Level 2%. 1123456789101112152025304050100 1 2