QBUS5001 (2021 S1): Final Exam Formulas
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FORMULAE SHEET
Probability
P(A|B) = P(A ∩ B)P(B) P(A ∪ B) = P(A) + P(B) − P(A ∩ B) P(Ai|B) = P(Ai)(B|Ai)∑ P�Aj�P�B�Aj�nj=1
Laws of Expected Value, Variance, CV, Covariance and Coefficient of Correlation
E[Xr] = ∑ xrP(X = x)x , r = 1,2,… [for a discrete random variable X] V[X] = E[X2] − (E[X])2 CV = SD[X]
E[X] E[aX ± b] = aE[X] ± b V[aX ± b] = a2V[X] Corr[X, Y] = Cov[X,Y]
SD[X]×SD[Y] E[aX ± bY] = aE[X] ± bE[Y] V[aX ± bY] = a2V[X] + b2V[Y] ± 2abCov[X, Y]
Sample Statistics
x̄ = ∑ xini=1
n
s2 = ∑ (xi−x̄)2ni=1
n−1
= ∑ xi2−nx̄2ni=1
n−1
CV = s
x̄

Outliers: Values lie outside the interval (Q1 - 1.5 × IQR, Q3 + 1.5 × IQR)
Probability Distributions
Distribution PMF/PDF, mean and variance X~Bin(n, p) P(X = x) = Cxnpx(1 − p)n−x, x = 0,1, … , n where Cxn = n!x!(n−x)! E[X] = np V[X] = np(1 − p) Bin(n, p) ≈ N�np, np(1 − p)� , continuity correction of 0.5, e.g. P(X = 10) ≈ P(10 − 0.5 ≤ X ≤ 10 + 0.5) Bin(n, p) ≈ Poi(np)
X~Poi(μ) P(X = x) = μxe−μx! , x = 0,1,2, … E[X] = μ V[X] = μ X~Exp(λ) f(x) = λe−λx, x ≥ 0, and F(x) = P(X ≤ x) = 1 − e−λx E[X] = 1
λ
V[X] = 1
λ2
X~N(μ,σ2) E[X] = μ V[X] = σ2
QBUS5001 (2021 S1): Final Exam Formulas
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Distribution Theory and Central Limit Theorem E[X] = μ; E[X̄] = μ; V[X] = σ2; V[X̄] = σ2
n

If X is normal, X̄~N �μ, σ2
n
�
If X is not normal, X̄ ≈ N �μ, σ2
n
�
approximately
Z = X̄ − μσ
√n ~N(0,1) p� ≈ N �p, p(1−p)n � approximately

Inference – Estimation
Cases Confidence Interval Estimates
Estimating μ when σ is known x̄ ± zα 2⁄ σ
√n
Estimating μ when σ is unknown x̄ ± tα 2⁄ ,n−1 s
√n
Estimating p p� ± zα 2⁄ �p�(1 − p�)n
Determining n for estimating μ with known
σ and error bound B n = �zα 2⁄ σB �2
Determining n for estimating μ with
unknown σ, known range R and error bound B n = �zα 2⁄ σB �2 where σ ≈ R4
Determining n for estimating p and error
bound B n = 14 �zα/2B �2 or n = p�q� �zα/2B �2
Determining n1 = n2 = n for estimating p1 − p2 and error bound B n = (p�1q�1 + p�2q�2) �zα2B�2 or n = 12 �zα2B�2

Inference – Hypothesis Testing
Cases Test Statistics under H0 Testing H0: μ = μ0 when σ is known Z = x̄−μ0σ
√n
Testing H0: μ = μ0 when σ is unknown t = x̄ − μ0s
√n Testing H0: p = p0 Z = p� − p0
�p0q0)n Testing H0: μ1 − μ2 = d when σ1 and σ2 are known. Z = x̄1 − x̄2 − d
�σ1
2n1 + σ22n2
QBUS5001 (2021 S1): Final Exam Formulas
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Testing H0: μ1 − μ2 = d when σ1 and σ2 are unknown but equal t = x̄1−x̄2−d�sp2� 1n1+ 1n2� where sp2 = (n1−1)s12+(n2−1)s22n1+n2−2 and df = n1 + n2 – 2 Testing H0: μ1 − μ2 = d when σ1 and σ2 are unknown and unequal t = x̄1 − x̄2 − d
��
s12n1 + s22n2� and df =
�
s12n1 + s22n2�2
�
s12n1�2n1 − 1 + �s2
2n2�2n2 − 1

Testing H0: p1 − p2 = 0 Z = p�1−p�2
�p�q��
1
n1
+
1
n2
�
where p� = X1+X2
n1+n2


F Test for the equality of two population variances Testing H0: σ12 = σ22 F = s12
s2
2 and df = (df1, df2)

Linear Regression Coefficient of determination, R2 R2 = SSRSST = 1 − SSESST For simple linear regression, the confidence interval for E(Y|X = xg) y� ± tα/2,n−2se�1n + �xg − x��2(n − 1)sx2 For simple linear regression, the prediction interval for ynew at X = xg y� ± tα/2,n−2se�1 + 1n + �xg − x��2(n − 1)sx2 Testing overall significance H0: β1 = β2 = . . . = βk = 0 F = MSRMSE ~ Fk,n−k−1 Testing individual regression coefficients βj, H0: βj = 0 vs H1: βj≠ 0 t = β�jSE(β�j) ~ tn−k−1 Adjusted R2 Get this value from Excel output Estimate of a regression coefficient Get this value from Excel output Standard error of the estimate of a regression coefficient Get this value from Excel output Predicted y-values, residuals and standardised residuals Get these values from Excel output

QBUS5001 (2021 S1): Final Exam Formulas
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Critical Values of Statistical Distributions – Must be obtained using Excel
norm.inv(1-α,0,1) or norm.s.inv(1-α) for zα t.inv(1-α,df) for tα,df

Some Excel Commands binom.dist(x,n,p,0) Compute PMF of the Bin(n,p) distribution binom.dist(x,n,p,1) Compute CDF of the Bin(n,p) distribution poisson.dist(x,μ,0) Compute PMF of the Poi(μ) distribution poisson.dist(x,μ,1) Compute CDF of the Poi(μ) distribution expon.dist(x,λ,1) Compute CDF of the Exp(λ) distribution norm.dist(x,μ,σ,1) Compute CDF of the N(μ,σ2) distribution t.dist(x,df,1) Compute CDF of the tdf(0,1) distribution f.dist(x,df1,df2,1) Compute CDF of the Fdf1,df2 distribution norm.inv(a,μ,σ) Find x such that P(X ≤ x) = a where X ~ N(µ,σ2) norm.inv(a,0,1) or norm.s.inv(a) Find z such that P(Z ≤ z) = a where Z ~ N(0,1) t.inv(a,df) Find x such that P(X ≤ x) = a where X ~ tdf (0,1) t.inv.2t(a,df) Find x such that P(X ≤ -x)+ P(X > x) = a, X ~ tdf (0,1) f.inv(a,df1,df2) Find x such that P(X ≤ x) = a where X ~ Fdf1,df2 combin(n,x) Compute the value of Cxn average(array) Compute the mean stdev.s(array) Compute the sample standard deviation quartile.exc(array, 1) Compute the first quartile quartile.exc(array, 3) Compute the third quartile percentile.exc(array, p) Compute the 100pth percentile correl(array1, array2) Compute the coefficient of correlation






































































































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