Semester One Examinations, 2024 STAT1201 Formulas Basics 1 00 1 n-1 >(xj - x)2 . Expected Values E(X)=>P(X=x)x, Var(X)=>P(X=x)(x-E(X))2, sd(X) = VVar(X), x x E(aX+b)=aE(X)+b, sd(aX+b)=|a|sd(X), E(X1+X2)=E(X1)+E(X2), if independent: Var(X1 + X2) = Var(X1) + Var(X2). Standardising If X ~ Normal(u, o), then Z = X- M ? ~ Normal(0, 1). Binomial Distributions E(X)=np, sd(X)=Vnp(1-p), p == , E(p)=p, sd(p) = n p(1 -p) . Test and Confidence Intervals based on Standard Errors estimate - hypothesised , estimate # t x se(estimate), t = se(estimate) se(x) = To se(x1-T2) = 1 n1 n2 ,2 SÍ s2 - +- , se(r) = V 1- 2 n - 2' se(p) = 1 p(1 - p) n . n2 , se(p1- P2) = 1 n1 P1(1 - p1) + P2(1 - p2) Confidence Intervals for Correlation p 1 2 1 - r = arctanh(r), r = tanh(z) = ez - e-z ez + e-z. approx. 1 (1=2In (+p). 1 Vn - 3 . Page 16 of 18
Semester One Examinations, 2024 STAT1201 Interpreting Strength of Evidence from a p-value Strong Moderate Weak Inconclusive 0 0.01 0.05 0.1 1 Pooled t-Test and Analysis of Variance ,2 Sp n1+ 22-2 (n1- 1)s2 + (n2-1)s2 , DFT= n-1, DFG = k -1, MS = DF' SS R2 = SST' SSG MSG MSR' Sp = VMSR, F = a - 0.05. Linear Regression y = bo + b1x, y = bo + b1x1+ b2x2, X2 1 0, if Group A. [ 1, if Group B, Odds and Logistic Regression odds = 1 - p' p OR = odds for group A odds for group B , In ( 1 -p p - = b0 + b1x. Chi-Squared Tests overall total (row total) x (column total) , ? expected (observed - expected)2 expected , df = (#rows - 1) x (#columns - 1). Signed-Rank Test E(X) = sd(S) = 1 24 n(n+1)(2n+1) , n(n+1) 4 . Rank-Sum Test E(W) = n1 (n1 + 2+1) , sd(W) = 2 n?n2(n1+n2 + 1) 12 . Page 17 of 18
Semester One Examinations, 2024 STAT1201 R Output You may find some of the following output from R to be useful in answering the questions on this examination paper. > pnorm(-0.9579) [1] 0.1690566 > pnorm(1.8059) [1] 0.964533 > pnorm(1.9845) [1] 0.9763999 > qnorm(0.975) [1] 1.959964 > pt(-2.2464, df = 4) [1] 0.04399771 > pt(-2.2464, df = 5) [1] 0.03730653 > pt(1.0562, df = 4) [1] 0.8247772 > pt(5.39, df = 343, lower.tail = FALSE) [1] 6.565767e-08 > pt(5.39, df = 344, lower.tail = FALSE) [1] 6.554509e-08 > pt(5.39, df = 345, lower.tail = FALSE) [1] 6.543332e-08 > pchisq(q = 151.8, df = 3, lower.tail = FALSE) [1] 1.077601e-32 > pchisq(q = 12.7625, df = 3, lower.tail = FALSE) [1] 0.005179404 > pchisq(q = 12.7625, df = 8, lower.tail = FALSE) [1] 0.1202867 > pf(22.64, df1 = 2, df2 = 30, lower.tail = FALSE) [1] 1.015371e-06 > pf(22.64, df1 = 2, df2 = 35, lower.tail = FALSE) [1] 4.904129e-07 > pf(21.20, df1 =