Numerical Descriptive Measures

1. Population Variance = σ 2. Sample Variance = s 2 =

2

=

∑iN 1 ( x i − μ ) =

2

N n ( x − x) 2 ∑i =1 i

n −1

3. Inter‐quartile Range = Q 3 − Q1

Expectation and variance 1. Expected value of X: μ = E ( X ) = ∑ xp ( x ) 2. Variance of X: σ

2 2 = ∑ ( x − μ ) p( x)

Probability 1. Additive Rule: P ( A ∪ B ) = P ( A) + P ( B ) − P ( A ∩ B ) 2. Multiplicative Rule: P ( A ∩ B ) = P ( A) P ( B ) , if A and B are independent 3. Complement Rule: P ( A) = 1 − P ( A)

Conditional Probability

1. Definition: P ( A | B ) =

P( A ∩ B) P( B)

2. Multiplicative Rule: P ( A ∩ B ) = P ( A) P ( B | A) = P ( B ) P ( A | B )

Binomial Distribution X ~ B(n, p)

1. P ( X = k ) = ⎜ ⎟ p (1 − p ) 2. E(X) = np; V(X) = np(1−p)

⎛n⎞ ⎝k⎠

k

n−k

=

n! k ! ( n − k )!

k n−k p (1 − p )

Normal Distribution X~ N(μ, σ)

1. Standard normal: Z =

X −μ

σ

Confidence Interval 1. z‐confidence interval:

2. t‐confidence interval: 3. Confidence interval for proportion: x ± zα / 2

x ± t α / 2 , df

σ

n

s ( df = n − 1)

∧ p ± zα / 2

n ∧ ∧ p (1 − p )

n

1

Sample size

sample size to estimate the parameter μ to within B units with (1‐α)100% confidence: n =

⎡ zα / 2σ ⎤ ⎢ B ⎥ ⎣ ⎦ X −μ σ n

2

Test statistics for µ and p

1. z‐test for µ : z= t-test for

2.

μ :t =

X −μ (d.f. = n−1) s n

p− p pq / n

∧

3. z-test for p : z =

,

np ≥ 5 and nq ≥ 5 (where q = 1 – p)

Test statistics for μ1 − μ 2 and p1 − p2 1. z‐test for μ1 − μ 2 : 1

( − ) − (μ − μ z= X X

2 1

2

σ +σ

2 1

2 2

)

( − ) − (μ − μ t= X X 1 ⎞ ⎛ 1 ⎜ + s ⎝n n ⎟ ⎠ 1 2 1 2 p 1 2 2 2 2

n

1

n

2

2. t‐test for μ1 − μ 2 when σ 1 , σ 2 unknown and σ 1 = σ 2 :

) ,

where d.f. =

n +n

1

2

2 −2 and S p =

(n1 − 1) s + (n2 − 1) s n1 + n2 − 2

2 1

3. t‐test for

μ

D

(for matched pairs): t =

X S

D D

− μD

, where d.f. = nD − 1

n

D

4. z‐test for p1 − p2 : z =

( p 1 − p 2 ) − ( p1 − p 2 ) 1 1 p q( + ) n1 n2

∧

∧ ∧

∧

∧

(where H0: p1 – p2 = 0 and p1 = ∧ ∧ ∧ ∧

∧ ∧ ∧ ∧ ∧ X1 ∧ X ∧ X + X2 ∧ ; p2 = 2 ; p = 1 ; q1 = 1 − p1 ; q 2 = 1 − p 2 ; q = 1 − p , n1 n2 n1 + n2

and all of n1 p 1 , n1 q 1 , n 2 p 2 , n 2 q 2 ≥ 5 )

2

Simple linear regression and correlation.

SS x = ∑ x

2 i

( ∑x ) −

i

2

SS y = ∑ y −

2 i

n ( ∑ yi

= ( ∑ xi ) − nx

2

2

)

2

SS xy = ∑ x i y −

i

n (∑ x i )(∑ y )

i

= ( ∑ yi ) − n y

2

2

n

= ∑ x i y − nx y

i

β1 =

∧

∧

SS xy SS x

∧ ∧

β0 = y − β 1 x

ei = y i − y i

2 SS xy

SSE = ∑ ei2 = SS y −

sε = SSE n−2 sε

SS x

S∧ =

β1

∧

SS x

t= r=

β1 − β1

s∧

β1

with d.f.= n‐2

SS xy SS x SS y

2 SS xy

R2 =

∧

SS x SS y

∧ ∧

=

SS y − SSE SS y

=

SSR SS y

y = β 0 + β1 x

1 $ ±t y α 2,n − 2 Sε 1 + + n

(x

g

−x

)

2

SS x

$ ±t y α 2,n − 2 Sε

xg − x 1 + n SS x

(

)

2

3