Eksamenssett.no
Ressurser
Skolenytt
Hoderegning
MET2
Formelark
Statistikk for økonomer
eksamenssett.no
Deskriptiv statistikk
•
x
ˉ
=
1
n
∑
i
=
1
n
x
i
\displaystyle \bar{x} = \frac{1}{n}\sum_{i=1}^{n} x_i
x
ˉ
=
n
1
i
=
1
∑
n
x
i
(gjennomsnitt)
•
s
2
=
1
n
−
1
∑
(
x
i
−
x
ˉ
)
2
\displaystyle s^2 = \frac{1}{n-1}\sum(x_i - \bar{x})^2
s
2
=
n
−
1
1
∑
(
x
i
−
x
ˉ
)
2
(utvalgsvarians)
•
C
V
=
s
/
x
ˉ
⋅
100
%
CV = s/\bar{x} \cdot 100\,\%
C
V
=
s
/
x
ˉ
⋅
100
%
(variasjonskoeffisient)
•
I
Q
R
=
Q
3
−
Q
1
IQR = Q_3 - Q_1
I
QR
=
Q
3
−
Q
1
(kvartilbredde)
Sannsynlighetsregning
•
P
(
A
∪
B
)
=
P
(
A
)
+
P
(
B
)
−
P
(
A
∩
B
)
P(A \cup B) = P(A) + P(B) - P(A \cap B)
P
(
A
∪
B
)
=
P
(
A
)
+
P
(
B
)
−
P
(
A
∩
B
)
•
P
(
A
∣
B
)
=
P
(
A
∩
B
)
/
P
(
B
)
P(A|B) = P(A \cap B)/P(B)
P
(
A
∣
B
)
=
P
(
A
∩
B
)
/
P
(
B
)
•
Bayes:
P
(
B
j
∣
A
)
=
P
(
A
∣
B
j
)
P
(
B
j
)
∑
P
(
A
∣
B
i
)
P
(
B
i
)
\displaystyle P(B_j|A) = \frac{P(A|B_j)P(B_j)}{\sum P(A|B_i)P(B_i)}
P
(
B
j
∣
A
)
=
∑
P
(
A
∣
B
i
)
P
(
B
i
)
P
(
A
∣
B
j
)
P
(
B
j
)
•
Uavhengighet:
P
(
A
∩
B
)
=
P
(
A
)
⋅
P
(
B
)
P(A \cap B) = P(A) \cdot P(B)
P
(
A
∩
B
)
=
P
(
A
)
⋅
P
(
B
)
Sannsynlighetsfordelinger
•
Binomisk:
P
(
X
=
k
)
=
(
n
k
)
p
k
(
1
−
p
)
n
−
k
P(X=k) = \binom{n}{k}p^k(1-p)^{n-k}
P
(
X
=
k
)
=
(
k
n
)
p
k
(
1
−
p
)
n
−
k
,
E
=
n
p
E = np
E
=
n
p
,
Var
=
n
p
(
1
−
p
)
\text{Var} = np(1-p)
Var
=
n
p
(
1
−
p
)
•
Poisson:
P
(
X
=
k
)
=
λ
k
e
−
λ
/
k
!
P(X=k) = \lambda^k e^{-\lambda}/k!
P
(
X
=
k
)
=
λ
k
e
−
λ
/
k
!
,
E
=
Var
=
λ
E = \text{Var} = \lambda
E
=
Var
=
λ
•
Normal:
Z
=
(
X
−
μ
)
/
σ
Z = (X - \mu)/\sigma
Z
=
(
X
−
μ
)
/
σ
•
E
(
a
X
+
b
)
=
a
E
(
X
)
+
b
E(aX+b) = aE(X)+b
E
(
a
X
+
b
)
=
a
E
(
X
)
+
b
,
Var
(
a
X
+
b
)
=
a
2
Var
(
X
)
\text{Var}(aX+b) = a^2\text{Var}(X)
Var
(
a
X
+
b
)
=
a
2
Var
(
X
)
Estimering og konfidensintervall
•
S
E
(
X
ˉ
)
=
s
/
n
SE(\bar{X}) = s/\sqrt{n}
SE
(
X
ˉ
)
=
s
/
n
(standardfeil)
•
z-intervall:
x
ˉ
±
z
α
/
2
⋅
σ
/
n
\bar{x} \pm z_{\alpha/2} \cdot \sigma/\sqrt{n}
x
ˉ
±
z
α
/2
⋅
σ
/
n
•
t-intervall:
x
ˉ
±
t
α
/
2
,
n
−
1
⋅
s
/
n
\bar{x} \pm t_{\alpha/2,n-1} \cdot s/\sqrt{n}
x
ˉ
±
t
α
/2
,
n
−
1
⋅
s
/
n
•
Proporsjon:
p
^
±
z
α
/
2
p
^
(
1
−
p
^
)
/
n
\hat{p} \pm z_{\alpha/2}\sqrt{\hat{p}(1-\hat{p})/n}
p
^
±
z
α
/2
p
^
(
1
−
p
^
)
/
n
•
Utvalgsstørrelse:
n
=
(
z
α
/
2
⋅
σ
/
E
)
2
n = (z_{\alpha/2} \cdot \sigma / E)^2
n
=
(
z
α
/2
⋅
σ
/
E
)
2
Hypotesetesting
•
t-test:
t
=
(
x
ˉ
−
μ
0
)
/
(
s
/
n
)
t = (\bar{x} - \mu_0)/(s/\sqrt{n})
t
=
(
x
ˉ
−
μ
0
)
/
(
s
/
n
)
,
d
f
=
n
−
1
df = n-1
df
=
n
−
1
•
z-test proporsjon:
z
=
(
p
^
−
p
0
)
/
p
0
(
1
−
p
0
)
/
n
z = (\hat{p}-p_0)/\sqrt{p_0(1-p_0)/n}
z
=
(
p
^
−
p
0
)
/
p
0
(
1
−
p
0
)
/
n
•
Forkast
H
0
H_0
H
0
hvis
p
p
p
-verdi
≤
α
\le \alpha
≤
α
•
Type I =
α
\alpha
α
, type II =
β
\beta
β
, styrke =
1
−
β
1-\beta
1
−
β
Regresjon
•
β
^
1
=
S
X
Y
/
S
X
X
\hat{\beta}_1 = S_{XY}/S_{XX}
β
^
1
=
S
X
Y
/
S
XX
,
β
^
0
=
Y
ˉ
−
β
^
1
X
ˉ
\hat{\beta}_0 = \bar{Y} - \hat{\beta}_1\bar{X}
β
^
0
=
Y
ˉ
−
β
^
1
X
ˉ
•
R
2
=
1
−
S
S
E
/
S
S
T
=
S
S
R
/
S
S
T
R^2 = 1 - SSE/SST = SSR/SST
R
2
=
1
−
SSE
/
SST
=
SSR
/
SST
•
R
ˉ
2
=
1
−
(
1
−
R
2
)
(
n
−
1
)
/
(
n
−
k
−
1
)
\bar{R}^2 = 1 - (1-R^2)(n-1)/(n-k-1)
R
ˉ
2
=
1
−
(
1
−
R
2
)
(
n
−
1
)
/
(
n
−
k
−
1
)
•
F-test:
F
=
M
S
R
/
M
S
E
F = MSR/MSE
F
=
MSR
/
MSE
,
d
f
1
=
k
df_1 = k
d
f
1
=
k
,
d
f
2
=
n
−
k
−
1
df_2 = n-k-1
d
f
2
=
n
−
k
−
1
•
t-test:
t
=
β
^
j
/
S
E
(
β
^
j
)
t = \hat{\beta}_j/SE(\hat{\beta}_j)
t
=
β
^
j
/
SE
(
β
^
j
)
•
s
e
=
S
S
E
/
(
n
−
k
−
1
)
s_e = \sqrt{SSE/(n-k-1)}
s
e
=
SSE
/
(
n
−
k
−
1
)
Modelldiagnostikk
•
Durbin-Watson:
D
W
≈
2
DW \approx 2
D
W
≈
2
= ingen autokorrelasjon
•
VIF
j
=
1
/
(
1
−
R
j
2
)
_j = 1/(1-R_j^2)
j
=
1/
(
1
−
R
j
2
)
, VIF
>
10
> 10
>
10
= problem
•
Cooks avstand:
D
i
>
1
D_i > 1
D
i
>
1
= innflytelsesrik
•
AIC/BIC: lavere = bedre modell