n m

m

X1, , Xn

σ2 µi = E [Xi ] i = 1, , n

Zi =Xi − µi

σi = 1, , n

Σi = 1

n

Z 2i = Σ

i = 1

n (Xi − µi )2

σ2

n

µi i = 1, , n

k

µ̂ i µi i = 1, , n

Σi = 1

n (Xi − µ̂ i )2

σ2

n − k

E [Xi ] = µ i = 1, , n

k = 1

X µ µi

Σi = 1

n (Xi − X )2

σ2

n − 1

Σi = 1

n (Xi − X )2

σ2=(N − 1 )S 2

σ2 ~ χ2n − 1

S 2

µi σ2

Xij ~ N (µi,σ2 ) i = 1, ,m j = 1, , n

H0 : µ1 = µ2 = = µm

H1 :

nm

Xij

Σi = 1

m

Σj = 1

n (Xij − µi )2

σ2 ~ χ2nm

i

Xi. =1nΣ

j = 1

n

Xij

Σi = 1

m

Σj = 1

n (Xij − Xi. )2

σ2 ~ χ2nm− m

SSW = Σi = 1

m

Σj = 1

n

(Xij − Xi. )2

SSW

E [SSW ]/σ2 = nm − m

E [SSW/ (nm − m )] = σ2

SSW/ (nm − m ) σ2

H0 m X1., , Xm.

µ

σ2/n

Xi. − µ√σ2/n

~ N (0,1 ) Σi = 1

m n (Xi. − µ )2

σ2 ~ χ2m

µ

µ

X.. =Σi = 1

m

Σj = 1

n

Xij

nm=Σi = 1

m

Xi.

m

H0

Σi = 1

m n (Xi. − X.. )2

σ2 ~ χ2m− 1

SSb = Σi = 1

m

n (Xi. − X.. )2

SSb

E [SSb ]/σ2 = m − 1

E [SSb/ (m − 1 )] = σ2

SSb/ (m − 1 ) σ2 H0

TS =SSb/ (m − 1 )

SSW/ (nm − m )

SSb/ (m − 1 ) σ2

H0

H0 TS

H0 SSb SSW

H0

TS ~ Fm− 1, nm−m

Fα,m− 1, nm−m 100 (1− α )

P Fm− 1, nm− m > Fα, m − 1, nm− m = α

α

H0 TS > Fα, m− 1, nm− m

H0

TS = v

p P Fm− 1, nm− m≥ v

H0 : µ1 = µ2 = µ3

H1 :

TS =SSb/ (m − 1 )

SSW/ (nm − m )

SSb = Σi = 1

m

n (Xi. − X.. )2

SSW = Σi = 1

m

Σj = 1

n

(Xij − Xi. )2

m = 3 n = 5

TS =SSb/ (m − 1 )

SSW/ (nm − m )=431.6667165.9667

= 2.6009

F0.05, 2, 12 = 3.89 H0

Σi = 1

m

Σj = 1

n

X 2ij = nmX 2

.. + SSb+ SSW

m

n1, , nm

n1 + + nm

Xij j = 1, , ni i = 1, ,m

Xij ~ N (µi, σ2 )

H0 : µ1 = µ2 = = µm

H1 :

H0

Σi = 1

m

Σj = 1

ni (Xij − µi )2

σ2 ~ χ2

Σi = 1

m

ni

SSW = Σi = 1

m

Σj = 1

ni

(Xij − Xi. )2

SSb = Σi = 1

m

ni (Xi. − X.. )2

TS =SSb/ (m − 1 )

SSW/ (Σi = 1

m

ni − m )

TS ~ Fm− 1,Σ

i= 1

m

ni − m

α

H0 TS > Fα, m− 1,Σ

i = 1

m

ni − m

H0

µ1, ,µm

T

α mC2

µi − µj i≠ j i, j = 1, ,m

1− α i≠ j

Xi. − Xj. − W < µi − µj < Xi. − Xj. +W

W =1√n

C (m,nm− m,α )

√SSW

nm− m

m = 3 n = 4

TS =SSb/2SSW/9

=0.36730.0431

= 8.5220

p P F2, 9≥ 8.5220 = 0.0046

X1. = 3.350 X2. = 3.350 X3. = 2.775

C (3, 9, 0.05 ) = 3.95

W=1√43.95

√0.0431 = 0.410

− 0.410 < µ1− µ2 < 0.410

0.165 < µ1− µ3 < 0.985

0.165 < µ2− µ3 < 0.985