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THE UNIVERSITY OF MELBOURNE Department of Economics ECON10005 QUANTITATIVE METHODS 1

FINAL EXAM FORMULA SHEET

Data Analysis x¯ =

Coefficient of variation

SAMPLE VARIANCE

sample Mean n

n

1 X xi n

s2 =

1 1 X (xi − x¯)2 = n−1 n−1 i=1

i=1

n

cov(X, Y ) = sxy COVariance

1X |xi − x¯| MAD = n i=1 Population: 1/n

i=1

n X

1 1 X (xi − x¯)(yi − y¯) = = n−1 n−1 i=1 n

n X

i=1

X pˆ = n

xi2 − n x¯2

!

cv = !

xi yi − n x¯ y¯

p Lp = (n + 1) 100

Proportion

r=

s x¯

sxy sx sy

Sample coefficient of correlation

percentile

Probability Theory Conditional probability

P [A|B] = Mean

Multiplication rule

P [A ∩ B] P [B]

E(X) = µ =

k X i=1

xi ·p(xi )

COV (X, Y ) = σxy =

n X m X i=1 j=1

If Y = a + b X, If X ∼ b(n, p): If Y = a + b X, If X ∼ N (µ, σ 2 ),

addition rule

P [A∩B] = P [A|B]·P [B]

2

V AR(X) = σ =

P [A∪B] = P [A]+P [B]−P [A∩B] k X i=1

(xi − µ)2 ·p(xi ) =

(xi − µx ) (yj −µy ) p(xi , yj ) =

E(Y ) = a + b E(X)

and

n X m X i=1 j=1

k X i=1

x2i ·p(xi ) − µ2

xi ·yj ·p(xi , yj ) −µx · µy

V AR(Y ) = b2 V AR(X )

binomial probability n! x n−x

P (X = x) = p(x) = E(Y ) = a + b E(X)

x!(n − x)!

and

p (1 − p)

n = number of trials; x = number of successes

V AR(Y ) = b2 V AR(X )

Z = (X − µ)/σ ∼ N (0, 1)

Continuous Distribution Densities Uniform: f (x) =

1 for a ≤ x ≤ b b−a

−(x−µ)2 1 Normal: f (x) = √ e 2σ2 −∞ < x < ∞ σ 2π

1

Sampling Distributions

For samples of size n:

¯ −µ X √ ∼ tn−1 s/ n

(n − 1)s2 2 ∼ χn−1 σ2

 p q pˆ ≃ N p, n

sampling distribution of proportion

Interval Estimators - Single Populations Estimator of mean when variance is known

¯ ± zα/2 √σ X n

¯ ± tα/2,n−1 √s X n

(n − 1)s2 (n − 1)s2 , 2 2 χα/2 χ1−α/2

!

n=

Tests - Single Populations ¯ −µ X √ σ/ n

t=

Estimating mean when variance is known

z

α/2

B

pˆ ± zα/2

r

n=



σ 2

sample size required to estimate mean with variance known

CI for chi squared

z=

Estimator of mean when variance is unknown

X¯ − µ √ s/ n

Estimator of proportion

√ 2 zα/2 pˆ qˆ B sample size required to estimate p

pˆ − p z=p pq/n

Estimating mean when variance is unknown

pˆ qˆ n

χ2 =

(n − 1)s2 σ2

Chi squared distribution

standardizing sample proportion

Interval Estimators - Two Populations mean for match pair

X¯D ±tα/2,nD −1

sD √ nD

ci for match pair

nD nD 1 X 1 X ¯ xDi (x1i −x2i ) = XD = nD nD i=1

i=1

¯ 1 − X¯2 ± zα/2 X

s

sD =

σ12 σ22 + n2 n1

sP

nD i=1 (xDi

¯ D )2 −X nD − 1

sd for match pair

pˆ1 − pˆ2 ± zα/2

r

pˆ1qˆ1 pˆ2qˆ2 + n2 n1

ci for independent when variance is known

Tests - Two Populations match pair if variance unknown

t=

¯D − µD X √ sD / nD

z=

¯ 2 ) − (µ1 − µ2 ) (X¯1 − X independent if q 2 σ22 σ1 variance known +n n1 and unequal 2

z= p1-p2 = D

(ˆ p1 − pˆ2 ) − (p1 − p2 ) q pˆ1 qˆ1 + pˆ2nqˆ22 n1 pˆ1 =

x1 n1

pˆ2 =

z=

x2 n2

2

(ˆ p1 − pˆ2 ) − (p1 − p2 ) r   p1-p2 pˆ qˆ n11 + n12

pˆ =

x1 + x2 n1 + n2

= 0

Simple Regression

βˆ0 = y¯ − βˆ1 x¯ s ˆβ0 = sε

sP

n i=1

Pn ¯)(yi − y¯) sxy i=1 (xi − x Pn = 2 βˆ1 = ¯)2 sx i=1 (xi − x

xi2/n (n − 1)sx2

s βˆ1 = p

sε

(n −

1)sx2

sε =

r

SSE n−2

test statistic

βˆj ± tα/2,n−2 · sβˆj

t=

βˆj − βj s βˆj

R2 =

2 sxy SSR SSE = 2 2 = 1− SST sx sy SST

n-2 d. f.

n X (yi −¯ y )2 SST = i=1

SSR =

n X i=1

(ˆ yi −¯ y)

2

 n n 2  X X sxy 2 2 2 e i = (n−1) sy − 2 (yi −ˆ yi ) = SSE = sx i=1 i=1

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