Statistics Formula Sheet

Statistics Formula Sheet Numerical Descriptive Measures 1. Population Variance = σ 2 = 2. Sample Variance = s 2 =

∑iN=1 ( x i − µ )

2

N 2 n ∑i =1 ( x i − x )

n −1 3. Inter-quartile Range = Q 3 − Q1

Expectation and variance µ = E ( X ) = ∑ xp ( x )

1. Expected value of X: 2. Variance of X: σ 2

= ∑ ( x − µ)

2

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 )

2. Multiplicative Rule:

P( B)

P ( A ∩ B ) = P ( A) P ( B | A) = P ( B ) P ( A | B )

Binomial Distribution X ~ B ( n, p ) 1.

 n  p k (1 − p ) n−k  k 

P( X = k ) = 

2. E(X) = np;

=

n! k! ( n − k )!

p

k

(1 − p )

n −k

V(X) = np(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

σ

x ± t α / 2 , df

∧ p ± zα / 2

n s n

( df = n − 1)

∧ ∧ p (1 − p ) n

Sample size sample size to estimate the parameter µ to within B units with (1-α )100% confidence: n =  zα / 2σ    B  

2

1

Test statistics for µ and p 1. z-test for µ : z=

X −µ σ n

X −µ (d.f. = n−1) s n µp − p 3. z-test for p : z = , np ≥ 5 and nq ≥ 5 (where q = 1 – p) qp n 2. t-test for µ : t =

Test statistics for µ1 − µ2 and p1 − p2

( − ) −( µ − µ z= X X σ +σ n n 1

1. z-test for µ1 − µ2 :

2

1

2

2

1

2

1

2

2

)

2. t-test for µ1 − µ2 when σ 1 , σ 2 unknown and σ 1 = σ 2 :

( − ) −( µ − µ t= X X 1   1 s  n + n  1

2

1

2

2

p

1

µ

D

,

2

where d.f. =

3. t-test for

)

n +n 1

2 2 −2 and S p =

(for matched pairs):

t=

X S

D

(n1 − 1) s12 + (n2 − 1) s22 n1 + n2 − 2

− µD

,

D

n

where d.f. =

nD − 1

D

∧

4. z-test for p1 − p2 :

z=

( p1 − p2 ) − ( p1 − p2 )

(where H0: p1 – p2 = 0 and and all of

∧

 1 1 µpq µ +   n1 n2 ∧

p1 =

   

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

n1 µp1 , n1 q$1 , n2 µp 2 , n2 q$2 ≥ 5 )

Simple linear regression and correlation.

2

SS x = ∑ x

2 i

( ∑x ) −

2

)

2

SS y = ∑ y − 2 i

= ( ∑ xi ) − nx

i

n ( ∑ yi

SS xy = ∑ x i y −

2

2

= ( ∑ yi ) − n y 2

n (∑ x i )(∑ y ) i

n

i

2

= ∑ x i y − nx y i

SS βµ1 = xy SS x β¶ 0 = y − βµ1 x e = y − µy i

i

i

SSE = ∑ ei2 = SS y −

SS xy2 SS x

SSE n−2 s S βµ = ε 1 SS x sε =

t=

βµ 1 − β1 with d.f.= n-2 sβµ 1

r= R2 =

SS xy SS x SS y SS xy2 SS x SS y

=

SS y − SSE SS y

SSR SS y

=

$ y = β¶ o + βµ1 x

(

$ y ± tα 2,n − 2 Sε

xg − x 1 1+ + n SS x

$ y ± tα 2,n − 2 Sε

xg − x 1 + n SS x

(

)

)

2

2

3