advanced statics

3 A Level in Statistics Population variance, σ 2 , = ∑x2 N − μ2 = N1 ∑(x − μ)2 Population standard deviation, σ, = ∑N x2 − μ2 = N1 ∑(x − μ)2 Sample variance, s 2 , = n 1− 1 2 (∑x)2 1 1 ∑(x − x)2 ∑x − n n − Sample standard deviation, s, = Binomial probability calculations: P(X = x) n x px (1 − p)n −x Binomial mean = np Binomial variance = np (1 − p) For a random sample of nx observations from N( μ , σ 2 ) 1 1 1 1 2 2 2 n x x n n x x − − ( ) = − ( − ) ∑ ∑ ∑ Pearson Edexcel Level 3 Advanced Subsidiary and Advanced GCE in Statistics 5Statistical Formulae and Tables – Issue 1 – August 2017 – © Pearson Education Limited 2017 ~ N(0, 1) Test statistic for a binomial proportion using normal distribution: pˆ − p ~ N(0, 1) p(1 − p) n Coefficients for least squares regression line: least squares regression line of y on x is y = a + bx , where a = y − bx S ∑(x − x)(y − y) the regression coefficient of y on x is b Bayes’ theorem for up to three events: P(A ) × P(B A ) PAj B= n j j ∑P(Ai ) × P(B Ai ) i=1 The Poisson probability formula: P(X = x) = e−λ λxx ! Poisson mean = λ Poisson variance = λ The exponential cumulative probability formula: X μ σ n −  Product moment correlation coefficient: r S S S x x y y x x y y x y x y xy xx yy i i i i i i i = × = ( − )( − ) { ( − )}{ ( − )} = − ( ) ∑ ∑ ∑ ∑ 2 2 i i i i n x x n y y n ∑ ∑ ∑ ∑ ∑ ∑ ( ) − ( ) − ( ) 2 2 2 2 ) ( 6 Pearson Edexcel Level 3 Advanced Subsidiary and Advanced GCE in Statistics Statistical Formulae and Tables – Issue 1 – August 2017 – © Pearson Education Limited 2017 P(X x) = 1 − e −λx Exponential mean = Exponential variance = E(aX ± bY) = aE(X) ± bE(Y) Var(aX ± bY) = a 2 Var(X) + b 2 Var(Y), for independent variables X and Y For a random sample of nx observations from N (μ, σ 2 ) X − μ S ~ tn–1 (also valid in matched-pairs situations) n For a random sample of nx observations from N (μx, σx 2 ) and, independently, a random sample of ny observations from N (μy , σy 2 ) For a random sample of nx observations from N (μx, σx 2 ) and, independently, a random sample of ny observations from N (μy , σy 2 ) where σx 2 = σy 2 = σ 2 (unknown) X − Y ) − (μx − μy ) ~ t nx+ny–2 where 1 + 1 Sp2 nxny 2 (nx − 1)Sx2 + (ny − 1)Sy 2 Sp = nx + ny − 2 Test statistic for the difference in two binomial proportions: p1 − p2 1 1 standard error where standard error = p × (1 − p) n1 + n2 ( ( ) ( ) 2 2 x y x y x y X Y μ μ σ σ n n − − − + ~ N(0, 1) Pearson Edexcel Level 3 Advanced Subsidiary and Advanced GCE in Statistics 7Statistical Formulae and Tables – Issue 1 – August 2017 – © Pearson Education Limited 2017 p 1 × n 1 + p 2 × n where p = 2 n1 + n2 Test for association and goodness of fit test: ∑(O i − E i )2 is approximately distributed as χ 2 Ei Analysis of variance (one-way and two-way): one-factor model xij = μ + αi + εij , where εij ~ N (0, σ 2 ) total sum of squares SS T i j x i Tn 2 T i2 − T 2 between groups sum of squares SS B = i ni n two-factor model (with m rows and n columns) xij = μ + αi + βj + εij , where εij ~ N (0, σ 2 ) total sum of squares SS T x i mn T2 i j between rows sum of squares SS R n mn T2 i between columns sum of squares SS C m mn T2 j Cohen’s d formula: x1 − x ) d =2