BUSS1020 In-semester Examination Formula Sheet Range Range= X largest Xsmallest Z-score Z = X Xˉ S Population Variance σ2 = N∑ i=1 ( Xi μ )2 N Population Standard Deviation σ= √√√√√√ N∑ i=1 ( Xi μ )2 N Sample Variance S2 = n∑ i=1 ( Xi Xˉ )2 n 1 Sample Standard Deviation S = √√√√√ n∑ i=1 ( Xi Xˉ )2 n 1 Coefficient of Variation CV = ( S Xˉ ) ·100% Sample Mean Xˉ = n∑ i=1 Xi n = X1+X2+X3+ …+Xn n Sample Covariance Cov(X ,Y )= n∑ i=1 ( Xi Xˉ )( Yi Yˉ ) n 1 Empirical Rule If distribution is bell-shaped, then approximately: 68% of data is within±1 standard deviation of the mean. 95% of data is within±2 standard deviation of the mean. 99.7% of data is within ±3 standard deviation of the mean. Chebyshev’s Rule Regardless of how the data is distributed, at least: ( 1 1 k2 ) ×100%, k > 1 will fall within k standard deviations of the mean. Counting Rules Multiplication rule: (k1)(k2)…(kn) Repetition rule: kn Factorial: n!= (n)(n 1)(n 2)…(2)(1) Permutations: nPk = n! (n k)! Combinations: nCk = n! k !(n k)! Sample Coefficient of Correlation r = Cov(X ,Y ) SX SY General Addition Rule P (A or B)= P (A)+P (B) P (A and B) Page 1 of 3 BUSS1020 In-semester Examination Conditional Probability P (A|B)= P (A and B) P (B) Independence A and B are independent if: P (A|B)= P (A) Marginal Probability P (A)= k∑ i=1 P (A|Bi )P (Bi ) = P (A|B1)P (B1)+P (A|B2)P (B2) +P (A|B3)P (B3)+ …+P (A|Bk)P (Bk) Bayes Theorem P (Bi |A)= P (A|Bi )P (Bi ) P (A) = P (A|Bi )P (Bi ) k∑ i=1 P (A|Bi )P (Bi ) = P (A|Bi )P (Bi ) P (A|B1)P (B1)+ …+P (A|Bk)P (Bk) Expected Value �
�= E(X )= N∑ i=1 xiP (X = xi ) Variance σ2 = E(X 2) [E(X )]2 = E [X E(X )]2 = N∑ i=1 [xi E(X )]2P (X = xi ) Covariance σXY = N∑ i=1 [xi E(X )] [ yi E(Y ) ] P (xi , yi ) = E(XY ) [E(X )][E(Y )] Binomial Distribution Formula P (X = x|n,pi)= nCxpix(1 pi)n x = n! x!(n x)!pi x(1 pi)n x Mean and Standard deviation: μ= npi σ= √ npi(1 pi) Poisson Distribution Formula P (X = x|λ)= e λλx x! Mean and Standard deviation: μ=λ σ= p λ Hypergeometric Distribution Formula P (X = x|n,N ,A)= ACx ·N ACn x NCn Mean and Standard deviation: μ= nA N σ= √ nA(N A) N2 · N n N 1 Sum of Two Random Variables: Expected Value E(X +Y )= E(X )+E(Y ) E(aX +bY )= E(aX )+E(bY )= aE(X )+bE(Y ) Sum of Two Random Variables: Variance Var (X +Y )=σ2X+Y =σ2X +σ2Y +2σX ,Y Var (aX +bY )=σ2aX+bY =σ2aX +σ2bY +2σaX ,bY = a2σ2X +b2σ2Y +2abσX ,Y Sum of Two Random Variables: Standard Deviation σX+Y = √ σ2X+Y Page 2 of 3 BUSS1020 In-semester Examination Z-score Z = X μ σ Assessing Normality IQR ～= 1.33S Uniform Distribution Formula P (X < x)= x a b a Mean and Standard deviation: μ= a+b 2 σ= √ (b a)2 12 Exponential Distribution Formula P (X< x)= 1 e λx Mean and Standard deviation: μ= 1 λ σ= 1 λ Standard Error of the Mean σXˉ = σp n Z-score for Sampling Distribution Z = Xˉ μσp n Standard Error of Population Proportion σp = √ pi(1 pi) n Z-score for Proportions Z = p pi√ pi(1 pi) n Page 3 of 3 Finite population correction factor = 1 Standard error of the sample mean for finite populations √ = 1 Standard error of the sample proportion for finite populations = (1 ) 1