Kiểm định giả thiết thống kê

Ch ’u ’ong 5
KI
’
ˆ
EM D
¯
I
.
NH GI
’
A THI
´
ˆ
ET TH
´
ˆ
ONG K
ˆ
E
1. C
´
AC KH
´
AI NI
ˆ
E
.
M
1.1 Gi
’
a thi
´
ˆet th
´
ˆong kˆe
Khi nghiˆen c
´
’
uu v
`
ˆe c´ac l
˜
inh v
’
u
.
c n`ao ¯d´o trong th
’
u
.
c t
´
ˆe ta th
’
u
`
’
ong ¯d
’
ua ra c´ac nhˆa
.
n x´et kh´ac
nhau v
`
ˆe c´ac ¯d
´
ˆoi t
’
u
’
o
.
ng quan tˆam. Nh
˜
’
ung nhˆa
.
n x´et nh
’
u vˆa
.
y th
’
u
`
’
ong ¯d
’
u
’
o
.
c coi l`a c´ac gi
’
a
thi
´
ˆet, ch´ung c´o th
’
ˆe ¯d´ung v`a c˜ung c´o th
’
ˆe sai. Viˆe
.
c sai ¯di
.
nh t´ınh ¯d´ung sai c
’
ua mˆo
.
t gi
’
a
thi
´
ˆet ¯d
’
u
’
o
.
c go
.
i l`a ki
’
ˆem ¯di
.
nh.
Gi
’
a s
’
’
u c
`
ˆan nghiˆen c
´
’
uu tham s
´
ˆo θ c
’
ua ¯da
.
i l
’
u
’
o
.
ng ng
˜
ˆau nhiˆen X, ng
’
u
`
’
oi ta ¯d
’
ua ra gi
’
a
thi
´
ˆet c
`
ˆan ki
’
ˆem ¯di
.
nh
H : θ = θ
0
Go
.
i H l`a gi
’
a thi
´
ˆet ¯d
´
ˆoi c
’
ua H th`ı H : θ = θ
0
.
T
`
’
u m
˜
ˆau ng
˜
ˆau nhiˆen W
X
= (X
1
, X
2
, . . . , X
n
) ta cho
.
n th
´
ˆong kˆe
ˆ
θ =
ˆ
θ(X
1
, X
2
, . . . , X
n
)
sao cho n
´
ˆeu H ¯d´ung th`ı
ˆ
θ c´o phˆan ph
´
ˆoi x´ac su
´
ˆat ho`an to`an x´ac ¯di
.
nh v`a v
´
’
oi m
˜
ˆau cu
.
th
’
ˆe
th`ı gi´a tri
.
c
’
ua
ˆ
θ s˜e t´ınh ¯d
’
u
’
o
.
c.
ˆ
θ ¯d
’
u
’
o
.
c go
.
i l`a tiˆeu chu
’
ˆan ki
’
ˆem ¯di
.
nh gi
’
a thi
´
ˆet H.
V
´
’
oi α b´e t`uy ´y cho tr
’
u
´
’
oc (α ∈ (0, 01; 0, 05)) ta t`ım ¯d
’
u
’
o
.
c mi
`
ˆen W
α
sao cho P (
ˆ
θ ∈
W
α
) = α.
W
α
¯d
’
u
’
o
.
c go
.
i l`a mi
`
ˆen b´ac b
’
o , α ¯d
’
u
’
o
.
c go
.
i l`a m
´
’
uc ´y ngh
˜
ia c
’
ua ki
’
ˆem ¯di
.
nh.
Th
’
u
.
c hiˆe
.
n ph´ep th
’
’
u ¯d
´
ˆoi v
´
’
oi m
˜
ˆau ng
˜
ˆau nhiˆen W
X
= (X
1
, X
2
, . . . , X
n
) ta ¯d
’
u
’
o
.
c m
˜
ˆau
cu
.
th
’
ˆe w
x
= (x
1
, x
2
, . . . , x
n
). T´ınh gi´a tri
.
c
’
ua
ˆ
θ ta
.
i w
x
= (x
1
, x
2
, . . . , x
n
) ta ¯d
’
u
’
o
.
c
θ
0
=
ˆ
θ(x
1
, x
2
, . . . , x
n
) (θ
0
¯d
’
u
’
o
.
c go
.
i l`a gi´a tri
.
quan s´at).
• N
´
ˆeu θ
0
∈ W
α
th`ı b´ac b
’
o gi
’
a thi
´
ˆet H v`a th
`
’
ua nhˆa
.
n gi
’
a thi
´
ˆet ¯d
´
ˆoi H.
• N
´
ˆeu θ
0
/∈ W
α
th`ı ch
´
ˆap nhˆa
.
n gi
’
a thi
´
ˆet H.
 Ch´u ´y
C´o tr
’
u
`
’
ong h
’
o
.
p gi
’
a thi
´
ˆet ki
’
ˆem ¯di
.
nh v`a gi
’
a thi
´
ˆet ¯d
´
ˆoi ¯d
’
u
’
o
.
c nˆeu cu
.
th
’
ˆe h
’
on. Ch
’
˘
ang ha
.
n:
H: θ ≤ θ
0
; H: θ > θ
0
Khi ¯d´o ta c´o ki
’
ˆem ¯di
.
nh mˆo
.
t ph´ıa.
85
86 Ch ’u ’ong 5. Ki
’
ˆem ¯di
.
nh gi
’
a thi
´
ˆet th
´
ˆong kˆe
1.2 Sai l
`
ˆam loa
.
i 1 v`a loa
.
i 2
Khi ki
’
ˆem ¯di
.
nh gi
’
a thi
´
ˆet th
´
ˆong kˆe, ta c´o th
’
ˆe m
´
˘
ac ph
’
ai mˆo
.
t trong hai loa
.
i sai l
`
ˆam sau:
i) Sai l
`
ˆam loa
.
i 1: l`a sai l
`
ˆam m
´
˘
ac ph
’
ai khi ta b´ac b
’
o mˆo
.
t gi
’
a thi
´
ˆet H trong khi H
¯d´ung.
X´ac su
´
ˆat m
´
˘
ac ph
’
ai sai l
`
ˆam loa
.
i 1 b
`
˘
ang P (
ˆ
θ ∈ W
α
) = α.
ii) Sai l
`
ˆam loa
.
i 2: l`a sai l
`
ˆam m
´
˘
ac ph
’
ai khi ta th
`
’
ua nhˆa
.
n gi
’
a thi
´
ˆet H trong khi H sai.
X´ac su
´
ˆat m
´
˘
ac ph
’
ai sai l
`
ˆam loa
.
i 2 b
`
˘
ang P (
ˆ
θ /∈ W
α
).
 Ch´u ´y
N
´
ˆeu ta mu
´
ˆon gi
’
am x´ac su
´
ˆat sai l
`
ˆam loa
.
i 1 th`ı s˜e l`am t
˘
ang x´ac su
´
ˆat sai l
`
ˆam loa
.
i 2 v`a
ng
’
u
’
o
.
c la
.
i.
D
¯
´
ˆoi v
´
’
oi mˆo
.
t tiˆeu chu
’
ˆan ki
’
ˆem ¯di
.
nh
ˆ
θ v`a v
´
’
oi m
´
’
uc ´y ngh
˜
ia α ta c´o th
’
ˆe t`ım ¯d
’
u
’
o
.
c vˆo s
´
ˆo
mi
`
ˆen b´ac b
’
o W
α
. Th
’
u
`
’
ong ng
’
u
`
’
oi ta
´
ˆan ¯di
.
nh tr
’
u
´
’
oc x´ac su
´
ˆat sai l
`
ˆam loa
.
i 1 (t
´
’
uc cho tr
’
u
´
’
oc
m
´
’
uc ´y ngh
˜
ia α) cho
.
n mi
`
ˆen b´ac b
’
o W
α
n`ao ¯d´o c´o x´ac su
´
ˆat sai l
`
ˆam loa
.
i 2 nh
’
o nh
´
ˆat.
2. KI
’
ˆ
EM D
¯
I
.
NH GI
’
A THI
´
ˆ
ET V
`
ˆ
E TRUNG B
`
INH
D
¯
a
.
i l
’
u
’
o
.
ng ng
˜
ˆau nhiˆen X c´o trung b`ınh E(X) = m ch
’
ua bi
´
ˆet. Ng
’
u
`
’
oi ta ¯d
’
ua ra gi
’
a
thi
´
ˆet
H : m = m
0
(H : m = m
0
)
2.1 Tr
’
u
`
’
ong h
’
o
.
p 1:

V ar(X) = σ
2
¯d˜a bi
´
ˆet
n ≥ 30 ho
˘
a
.
c (n < 30 v`a X c´o phˆan ph
´
ˆoi chu
’
ˆan)
Cho
.
n th
´
ˆong kˆe U =
(X − m
0
)
√
n
σ
. N
´
ˆeu H
0
¯d´ung th`ı U ∈ N(0, 1)
V
´
’
oi m
´
’
uc ´y ngh
˜
ia α cho tr
’
u
´
’
oc, x´ac ¯di
.
nh phˆan vi
.
chu
’
ˆan u
1−
α
2
. Ta t`ım ¯d
’
u
’
o
.
c mi
`
ˆen b´ac
b
’
o
W
α
= {u : |u| > u
1−
α
2
} = (−∞;−u
1−
α
2
) ∪ (u
1−
α
2
; +∞)
V`ı
P (U ∈ W
α
) = P (U < −u
1−
α
2
+ P (U > u
1−
α
2
)
= P (U < u
α
2
) + 1 − P (U > u
1−
α
2
)
=
α
2
+ 1 − (1 −
α
2
) = α
L
´
ˆay m
˜
ˆau cu
.
th
’
ˆe v`a t´ınh gi´a tri
.
quan s´at u
0
=
|x − m
0
|
σ
√
n .
So s´anh u
0
v`a u
1−
α
2
.
2. Ki
’
ˆem ¯di
.
nh gi
’
a thi
´
ˆet v
`
ˆe trung b`ınh 87
• N
´
ˆeu u
0
> u
1−
α
2
(u
0
∈ W
α
) th`ı b´ac b
’
o gi
’
a thi
´
ˆet H v`a ch
´
ˆap nhˆa
.
n H.
• N
´
ˆeu u
0
< u
1−
α
2
(u
0
/∈ W
α
) th`ı ch
´
ˆap nhˆa
.
n H
0
.
• V´ı du
.
1 Mˆo
.
t t´ın hiˆe
.
u c
’
ua gi´a tri
.
m ¯d
’
u
’
o
.
c g
’
’
oi t
`
’
u ¯di
.
a ¯di
’
ˆem A v`a ¯d
’
u
’
o
.
c nhˆa
.
n
’
’
o ¯di
.
a
¯di
’
ˆem B c´o phˆan ph
´
ˆoi chu
’
ˆan v
´
’
oi trung b`ınh m v`a ¯dˆo
.
lˆe
.
ch tiˆeu chu
’
ˆan σ = 2. Tin r
`
˘
ang
gi´a tri
.
c
’
ua t´ın hiˆe
.
u m = 8 ¯d
’
u
’
o
.
c g
’
’
oi m
˜
ˆoi ng`ay. Ng
’
u
`
’
oi ta ti
´
ˆen h`anh ki
’
ˆem tra gi
’
a thi
´
ˆet n`ay
b
`
˘
ang c´ach g
’
’
oi 5 t´ın hiˆe
.
u mˆo
.
t c´ach ¯dˆo
.
c lˆa
.
p trong ng`ay th`ı th
´
ˆay g´ıa tri
.
trung b`ınh nhˆa
.
n
¯d
’
u
’
o
.
c ta
.
i ¯di
.
a ¯di
’
ˆem B l`a X = 9, 5. V
´
’
oi ¯dˆo
.
tin cˆa
.
y 95%, h˜ay ki
’
ˆem tra gi
’
a thi
´
ˆet m = 8 ¯d´ung
hay khˆong?
Gi
’
ai
Ta c
`
ˆan ki
’
ˆem ¯di
.
nh gi
’
a thi
´
ˆet H : m
0
= 8 (H : m
0
= 8)
Ta c´o n = 5 < 30. D
¯
ˆo
.
tin cˆa
.
y 1 − α = 0, 95 =⇒ 1 −
α
2
= 0, 975
Phˆan vi
.
chu
’
ˆan u
0,975
= 1, 96.
Mi
`
ˆen b´ac b
’
o l`a W
α
= (−∞;−1, 96) ∪ (1, 96; +∞).
Gi´a tri
.
quan s´at u
0
=
|x − m
0
|
σ
√
n =
9, 5 − 8
2
√
5 = 1, 68.
Ta th
´
ˆay m
0
/∈ W
α
nˆen gi
’
a thi
´
ˆet H ¯d
’
u
’
o
.
c ch
´
ˆap nhˆa
.
n.
2.2 Tr
’
u
`
’
ong h
’
o
.
p 2:

σ
2
ch
’
ua bi
´
ˆet
n ≥ 30
Trong tr
’
u
`
’
ong h
’
o
.
p n`ay ta v
˜
ˆan cho
.
n th
´
ˆong kˆe nh
’
u trˆen trong ¯d´o ¯dˆo
.
lˆe
.
ch tiˆeu chu
’
ˆan σ
¯d
’
u
’
o
.
c thay b
’
’
oi ¯dˆo
.
lˆe
.
ch tiˆeu chu
’
ˆan c
’
ua m
˜
ˆau ng
˜
ˆau nhiˆen S

.
U =
(X − m
0
)
S

√
n
N
´
ˆeu H ¯d´ung th`ı U ∈ N(0, 1). T
’
u
’
ong t
’
u
.
nh
’
u trˆen ta c´o mi
`
ˆen b´ac b
’
o l`a
W
α
= {u : |u| > u
1−
α
2
} = (−∞; u
1−
α
2
) ∪ (u
1−
α
2
; +∞)
L
´
ˆay m
˜
ˆau cu
.
th
’
ˆe v`a ta t´ınh gi´a tri
.
quan s´at u
0
=
|x − m
0
|
s

√
n .
So s´anh u
0
v`a u
1−
α
2
.
• N
´
ˆeu u
0
> u
1−
α
2
(u
0
∈ W
α
) th`ı b´ac b
’
o gi
’
a thi
´
ˆet H v`a ch
´
ˆap nhˆa
.
n H.
• N
´
ˆeu u
0
< u
1−
α
2
(u
0
/∈ W
α
) th`ı ch
´
ˆap nhˆa
.
n H
0
.
88 Ch ’u ’ong 5. Ki
’
ˆem ¯di
.
nh gi
’
a thi
´
ˆet th
´
ˆong kˆe
• V´ı du
.
2 Mˆo
.
t nh´om nghiˆen c
´
’
uu tuyˆen b
´
ˆo r
`
˘
ang trung b`ınh mˆo
.
t ng
’
u
`
’
oi v`ao siˆeu thi
.
X
tiˆeu h
´
ˆet 140 ng`an ¯d
`
ˆong. Cho
.
n mˆo
.
t m
˜
ˆau ng
˜
ˆau nhiˆen g
`
ˆom 50 ng
’
u
`
’
oi mua h`ang, t´ınh ¯d
’
u
’
o
.
c
s
´
ˆo ti
`
ˆen trung b`ınh ho
.
tiˆeu l`a 154 ng`an ¯d
`
ˆong v
´
’
oi ¯dˆo
.
lˆe
.
ch tiˆeu chu
’
ˆan ¯di
`
ˆeu ch
’
inh c
’
ua m
˜
ˆau
l`a S

= 62. V
´
’
oi m
´
’
uc ´y ngh
˜
ia 0,02 h˜ay ki
’
ˆem ¯di
.
nh xem tuyˆen b
´
ˆo c
’
ua nh´om nghiˆen c
´
’
uu c´o
¯d´ung hay khˆong?
Gi
’
ai
Ta c
`
ˆan ki
’
ˆem ¯di
.
nh gi
’
a thi
´
ˆet H : m = 140 (H : m = 140)
Ta c´o n = 50 > 30 v`a 1 −
α
2
= 0, 99.
Phˆan v´ı chu
’
ˆan u
0,99
= 2, 33.
Mi
`
ˆen b´ac b
’
o W
α
= (−∞;−2, 33) ∪ (2, 33; +∞)
Gi´a tri
.
quan s´at u
0
=
|x − m
0
|
S

√
n =
154 − 140
62
√
50 = 1, 59.
Ta th
´
ˆay u
0
/∈ W
α
nˆen ch
’
ua c´o c
’
o s
’
’
o ¯d
’
ˆe loa
.
i b
’
o H. Ta
.
m th
`
’
oi ch
´
ˆap nhˆa
.
n r
`
˘
ang b´ao c´ao
c
’
ua nh´om nghiˆen c
´
’
uu l`a ¯d´ung.
2.3 Tr
’
u
`
’
ong h
’
o
.
p 3:

σ
2
ch
’
ua bi
´
ˆet
n < 30 v`a X c´o phˆan ph
´
ˆoi chu
’
ˆan
Cho
.
n th
´
ˆong kˆe
T =
(X − m
0
)
S

√
n
N
´
ˆeu H ¯d´ung th`ı T ∈ T (n − 1)
V
´
’
oi m
´
’
uc ´y ngh
˜
ia α cho tr
’
u
´
’
oc, ta x´ac ¯di
.
nh phˆan vi
.
Student (n − 1) bˆa
.
c t
’
u
.
do m
´
’
uc
1 −
α
2
l`a t
1−
α
2
.
Khi ¯d´o mi
`
ˆen b´ac b
’
o l`a
W
α
= {t : |t| > t
1−
α
2
} = (−∞;−t
1−
α
2
) ∪ (t
1−
α
2
; +∞)
L
´
ˆay m
˜
ˆau cu
.
th
’
ˆe v`a t´ınh gi´a tri
.
quan s´at t
0
=
|x − m
0
|
s

√
n .
• N
´
ˆeu t
0
> t
1−
α
2
(t
0
∈ W
α
) th`ı b´ac b
’
o gi
’
a thi
´
ˆet H v`a ch
´
ˆap nhˆa
.
n H.
• N
´
ˆeu t
0
< t
1−
α
2
(t
0
/∈ W
α
) th`ı ch
´
ˆap nhˆa
.
n H.
• V´ı du
.
3 Tro
.
ng l
’
u
’
o
.
ng c
’
ua c´ac bao ga
.
o l`a ¯da
.
i l
’
u
’
o
.
ng ng
˜
ˆau nhiˆen c´o phˆan ph
´
ˆoi chu
’
ˆan
v
´
’
oi tro
.
ng l
’
u
’
o
.
ng trung b`ınh l`a 50kg. Sau mˆo
.
t kho
’
ang th
`
’
oi gian hoa
.
t ¯dˆo
.
ng ng
’
u
`
’
oi ta nghi
ng
`
’
o tro
.
ng l
’
u
’
o
.
ng c´ac bao ga
.
o c´o thay ¯d
’
ˆoi. Cˆan 25 bao ga
.
o thu ¯d
’
u
’
o
.
c c´ac k
´
ˆet qu
’
a sau
3. Ki
’
ˆem ¯di
.
nh gi
’
a thi
´
ˆet v
`
ˆe t
’
y lˆe 89
X(kh
´
ˆoi l
’
u
’
o
.
ng) n
i
(s
´
ˆo bao)
48 − 48, 5 2
48, 5 − 49 5
49 − 49, 5 10
49, 5 − 50 6
50 − 50, 5 2
V
´
’
oi ¯dˆo
.
tin cˆa
.
y 99%, h˜ay k
´
ˆet luˆa
.
n v
`
ˆe ¯di
`
ˆeu nghi ng
`
’
o n´oi trˆen.
Gi
’
ai
X´et gi
’
a thi
´
ˆet H : m = 50
T =
(X − 50)
√
25
S

∈ T (24)
x
i
− x
i+1
x
0
i
n
i
(s
´
ˆo bao) u
i
n
i
x
2
i
n
i
48 − 48, 5 48,25 2 96,5 4656,125
48, 5 − 49 48,75 5 243,75 11882,812
49 − 49, 5 49,25 10 492,5 24255,625
49, 5 − 50 49,75 6 298,5 14850,375
50 − 50, 5 50,25 2 100,5 5050,125

25 1231,75 60695,062
Ta c´o 1 − α = 0, 99 =⇒ 1 −
α
2
= 0, 995
Phˆan vi
.
Student m
´
’
uc 0,995 v
´
’
oi 24 bˆa
.
c t
’
u
.
do l`a t
1−
α
2
= u
0,995
= 2, 797
Mi
`
ˆen b´ac b
’
o l`a W
α
= (−∞;−2, 797) ∪ (2, 797;∞)
x =
1231,75
25
= 49, 27.
s
2
=
60695,06
25
− (49, 27)
2
= 2427, 8 − 2427, 53 = 0, 27
s

2
=
25
24
0, 27 = 0, 2812 =⇒ s

= 0, 53
Gi´a tri
.
quan s´at t
0
=
|(49,27−50)|
√
25
0,53
= 6, 886
Ta th
´
ˆay t
0
∈ W
α
, nˆen gi
’
a thi
´
ˆet bi
.
b´ac b
’
o. Vˆa
.
y ¯di
`
ˆeu nghi ng
`
’
o l`a ¯d´ung.
3. KI
’
ˆ
EM D
¯
I
.
NH GI
’
A THI
´
ˆ
ET V
`
ˆ
E T
’
Y L
ˆ
E
.
Gi
’
a s
’
’
u t
’
ˆong th
’
ˆe c´o hai loa
.
i ph
`
ˆan t
’
’
u c´o t´ınh ch
´
ˆat A v`a khˆong c´o t´ınh ch
´
ˆat A, trong
¯d´o t
’
y lˆe
.
ph
`
ˆan t
’
’
u c´o t´ınh ch
´
ˆat A l`a p
0
ch
’
ua bi
´
ˆet. Ta ¯d
’
ua ra thi
´
ˆet
H : p = p
0
Lˆa
.
p m
˜
ˆau ng
˜
ˆau nhiˆen W
X
= (X
1
, X
2
, . . . , X
n
) v`a t´ınh t
’
y lˆe
.
f c´ac ph
`
ˆan t
’
’
u c
’
ua m
˜
ˆau c´o
t´ınh ch
´
ˆat A.
90 Ch ’u ’ong 5. Ki
’
ˆem ¯di
.
nh gi
’
a thi
´
ˆet th
´
ˆong kˆe
V
´
’
oi m
´
’
uc ´y ngh
˜
ia α cho tr
’
u
´
’
oc, x´ac ¯di
.
nh phˆan vi
.
chu
’
ˆan u
1−
α
2
. Mi
`
ˆen b´ac b
’
o l`a
W
α
= {u : |u| > u
1−
α
2
} = (−∞; u
1−
α
2
) ∪ (u
1−
α
2
; +∞)
L
´
ˆay m
˜
ˆau cu
.
th
’
ˆe v`a t´ınh gi´a tri
.
quan s´at u
0
=
|f − p
0
|
√
n
√
p
0
q
0
• N
´
ˆeu u
0
> u
1−
α
2
(u
0
∈ W
α
) th`ı b´ac b
’
o H v`a ch
´
ˆap nhˆa
.
n H.
• N
´
ˆeu u
0
< u
1−
α
2
(u
0
/∈ W
α
) th`ı ch
´
ˆap nhˆa
.
n H.
• V´ı du
.
4 T
’
y lˆe
.
ph
´
ˆe ph
’
ˆam
’
’
o mˆo
.
t nh`a m´ay c
`
ˆan ¯da
.
t l`a 10%. Sau khi c
’
ai ti
´
ˆen, ki
’
ˆem tra
400 s
’
an ph
’
ˆam th`ı th
´
ˆay c´o 32 ph
´
ˆe ph
’
ˆam v
´
’
oi ¯dˆo
.
tin cˆa
.
y 99%. H˜ay x´et xem viˆe
.
c c
’
ai ti
´
ˆen
k˜y thuˆa
.
t c´o k
´
ˆet qu
’
a hay khˆong?
Gi
’
ai
Ta c´o n = 400
Go
.
i p l`a t
’
y lˆe
.
ph
´
ˆe ph
’
ˆam c
’
ua nh`a m´ay .Ta ki
’
ˆem ¯di
.
nh gi
’
a thi
´
ˆet
H : p = 0, 1. (gi
’
a thi
´
ˆet ¯d
´
ˆoi H : p < 0, 1)
T
’
y lˆe
.
ph
´
ˆe ph
’
ˆam trong 400 s
’
an ph
’
ˆam l`a f =
32
400
= 0, 08
D
¯
ˆo
.
tin cˆa
.
y 1 − α = 0, 99 =⇒ 1 −
α
2
= 0, 995 =⇒ u
0,995
= 2, 576
Mi
`
ˆen b´ac b
’
o l`a W
α
= (−∞;−2, 576) ∪ (2, 576; +∞)
Gi´a tri
.
quan s´at u
0
=
(|0,08−0,1|)
√
400
√
0,1.0,9
= 1, 333 /∈ W
α
.
Do ¯d´o ch
´
ˆap nhˆa
.
n H
0
.
Vˆa
.
y viˆe
.
c c
’
ai ti
´
ˆen c´o hiˆe
.
u qu
’
a.
4. KI
’
ˆ
EM D
¯
I
.
NH GI
’
A THI
´
ˆ
ET V
`
ˆ
E PH
’
U
’
ONG SAI
Gi
’
a s
’
’
u X l`a ¯da
.
i l
’
u
’
o
.
ng ng
˜
ˆau nhiˆen c´o phˆan ph
´
ˆoi chu
’
ˆan v
´
’
oi ph
’
u
’
ong sai V ar(X) ch
’
ua
bi
´
ˆet. Ta ¯d
’
ua ra gi
’
a thi
´
ˆet
H : V ar(X) = σ
2
0
Lˆa
.
p m
˜
ˆau ng
˜
ˆau nhiˆen W
X
= (X
1
, X
2
, . . . , X
n
) v`a cho
.
n th
´
ˆong kˆe
χ
2
=
(n − 1)S

2
σ
2
0
N
´
ˆeu H ¯d´ung th`ı χ
2
c´o phˆan ph
´
ˆoi ” khi−b`ınh ph
’
u
’
ong ” v
´
’
oi n − 1 bˆa
.
c t
’
u
.
do.
V
´
’
oi m
´
’
uc ´y ngh
˜
ia α cho tr
’
u
´
’
oc, ta x´ac ¯di
.
nh c´ac phˆan vi
.
”khi−b`ınh ph
’
u
’
ong” χ
2
n−1,
α
2
, χ
2
n−1,1−
α
2
(n − 1) bˆa
.
c t
’
u
.
do, m
´
’
uc
α
2
, 1 −
α
2
. Khi ¯d´o mi
`
ˆen b´ac b
’
o l`a
5. Ki
’
ˆem ¯di
.
nh gi
’
a thi
´
ˆet m
.
ˆot ph´ıa 91
W
α
= {t : t < χ
2
n−1,
α
2
ho
˘
a
.
c t > χ
2
n−1,1−
α
2
} = (−∞; χ
2
n−1,
α
2
) ∪ (χ
2
n−1,1−
α
2
; +∞)
L
´
ˆay m
˜
ˆau cu
.
th
’
ˆe v`a t´ınh gi´a tri
.
quan s´at χ
2
0
=
(n − 1)s

2
σ
2
0
.
• N
´
ˆeu χ
2
0
< χ
2
n−1,
α
2
ho
˘
a
.
c χ
2
0
> χ
2
n−1,1−
α
2
(χ
2
0
∈ W
α
) th`ı b´ac b
’
o H v`a ch
´
ˆap nhˆa
.
n H.
• N
´
ˆeu χ
2
n−1,
α
2
< χ
2
0
< χ
2
n−1,1−
α
2
(χ
2
0
/∈ W
α
) th`ı ch
´
ˆap nhˆa
.
n H.
• V´ı du
.
5 N
´
ˆeu m´ay m´oc hoa
.
t ¯dˆo
.
ng b`ınh th
’
u
`
’
ong th`ı tro
.
ng l
’
u
’
o
.
ng c
’
ua s
’
an ph
’
ˆam l`a ¯da
.
i
l
’
u
’
o
.
ng ng
˜
ˆau nhiˆen X c´o phˆan ph
´
ˆoi chu
’
ˆan v
´
’
oi D(X) = 12. Nghi ng
`
’
o m´ay hoa
.
t ¯dˆo
.
ng khˆong
b`ınh th
’
u
`
’
ong ng
’
u
`
’
oi ta cˆan th
’
’
u 13 s
’
an ph
’
ˆam v`a t´ınh ¯d
’
u
’
o
.
c s

2
= 14, 6. V
´
’
oi m
´
’
uc ´y ngh
˜
ia
α = 0, 05. H˜ay k
´
ˆet luˆa
.
n ¯di
`
ˆeu nghi ng
`
’
o trˆen c´o ¯d´ung hay khˆong?
Gi
’
ai
Ta ki
’
ˆem ¯di
.
nh gi
’
a thi
´
ˆet H : V ar(X) = 12 ; H : V ar(X) = 12.
T
`
’
u c´ac s
´
ˆo liˆe
.
u c
’
ua b`ai to´an ta t`ım ¯d
’
u
’
o
.
c χ
2
0
=
(13−1)14,6
12
= 14, 6
V
´
’
oi α = 0, 05, tra b
’
ang phˆan vi
.
χ
2
v
´
’
oi (n − 1) = 12 bˆa
.
c t
’
u
.
do ta ¯d
’
u
’
o
.
c
χ
2
α
2
= χ
2
0,025
= 4, 4 v`a χ
2
1−
α
2
= χ
2
0,975
= 23, 3
Ta th
´
ˆay 4, 4 < 14, 6 < 23, 3 nˆen ch
´
ˆap nhˆa
.
n gi
’
a thi
´
ˆet H.
Vˆa
.
y ¯di
`
ˆeu nghi ng
`
’
o trˆen l`a khˆong ¯d´ung. M´ay v
˜
ˆan hoa
.
t ¯dˆo
.
ng b`ınh th
’
u
`
’
ong.
5. KI
’
ˆ
EM D
¯
I
.
NH M
ˆ
O
.
T PH
´
IA
Trong c´ac b`ai to´an trˆen ta ch
’
i x´et gi
’
a thi
´
ˆet ¯d
´
ˆoi c´o da
.
ng H : θ = θ
0
. Ta c˜ung c´o th
’
ˆe
gi
’
ai b`ai to´an ki
’
ˆem ¯di
.
nh v
´
’
oi gi
’
a thi
´
ˆet ¯d
´
ˆoi c´o da
.
ng: H : θ < θ
0
ho
˘
a
.
c H : θ > θ
0
. Khi gi
’
ai
c´ac b`ai to´an n`ay ta c˜ung ´ap du
.
ng c´ac qui t
´
˘
ac ¯d˜a ¯d
’
u
’
o
.
c tr`ınh b`ay v
´
’
oi ch´u ´y l`a:
i) Khi t´ınh g´ıa tri
.
quan s´at u
0
(ho
˘
a
.
c t
0
) trong c´ac qui t
´
˘
ac ki
’
ˆem ¯di
.
nh trˆen ta b
’
o d
´
ˆau
tri
.
tuyˆe
.
t ¯d
´
ˆoi
’
’
o t
’
’
u s
´
ˆo v`a thay b
`
˘
ang d
´
ˆau ngo
˘
a
.
c ¯d
’
on ( ). Ch
’
˘
ang ha
.
n u
0
=
(x − µ
0
)
σ
√
n.
ii) N
´
ˆeu gi
’
a thi
´
ˆet ¯d
´
ˆoi c´o da
.
ng H : θ > θ
0
th`ı ta so s´anh g´ıa tri
.
quan s´at u
0
v
´
’
oi
u
γ
= u
1−α
(ho
˘
a
.
c t
γ
= t
1−α
, ho
˘
a
.
c χ
2
1−α
).
N
´
ˆeu u
0
> u
γ
(ho
˘
a
.
c t
0
> t
γ
, χ
2
0
> χ
2
1−α
) th`ı b´ac b
’
o H v`a th
`
’
ua nhˆa
.
n H. N
´
ˆeu ng
’
u
’
o
.
c
la
.
i th`ı ch
´
ˆap nhˆa
.
n H.
iii) N
´
ˆeu gi
’
a thi
´
ˆet ¯d
´
ˆoi c´o da
.
ng H : θ < θ
0
th`ı ta so s´anh u
0
v
´
’
oi u
γ
= −u
1−α
, (ho
˘
a
.
c
t
γ
= −t
1−α
, ho
˘
a
.
c χ
2
α
).
N
´
ˆeu u
0
< −u
1−α
;(ho
˘
a
.
c t
0
< −t
1−α
, χ
2
0
< χ
2
α
) th`ı b´ac b
’
o H.N
´
ˆeu ng
’
u
’
o
.
c la
.
i th`ı ch
´
ˆap
nhˆa
.
n H.
92 Ch ’u ’ong 5. Ki
’
ˆem ¯di
.
nh gi
’
a thi
´
ˆet th
´
ˆong kˆe
• V´ı du
.
6 Mˆo
.
t nh`a s
’
an xu
´
ˆat thu
´
ˆoc ch
´
ˆong di
.
´
’
ung th
’
u
.
c ph
’
ˆam tuyˆen b
´
ˆo r
`
˘
ang 90% ng
’
u
`
’
oi
d`ung thu
´
ˆoc th
´
ˆay thu
´
ˆoc c´o t´ac du
.
ng trong v`ong 8 gi
`
’
o. Ki
’
ˆem tra 200 ng
’
u
`
’
oi bi
.
di
.
´
’
ung
th
’
u
.
c ph
’
ˆam th`ı th
´
ˆay trong v`ong 8 gi
`
’
o thu
´
ˆoc l`am gi
’
am b
´
’
ot di
.
´
’
ung ¯d
´
ˆoi v
´
’
oi 160 ng
’
u
`
’
oi. H˜ay
ki
’
ˆem ¯di
.
nh xem l
`
’
oi tuyˆen b
´
ˆo trˆen c
’
ua nh`a s
’
an xu
´
ˆat c´o ¯d´ung hay khˆong v
´
’
oi m
´
’
uc ´y ngh
˜
ia
α = 0, 01.
Gi
’
ai
Ta ¯d
’
ua ra gi
’
a thi
´
ˆet H : p
0
= 0, 9 (H < 0, 9)
α = 0, 01 −→ 1 − α = 0, 99 =⇒ −u
1−α
= −2, 326
f =
160
200
= 0, 8
u
0
=
f − p
0

p
0
(1 − p
0
)
√
n =
0, 8 − 0, 9
√
0, 9 × 0, 1
√
200 = −
0, 1
0, 3
.14, 14 = −4, 75
Ta th
´
ˆay u
0
< −u
1−α
nˆen b´ac b
’
o gi
’
a thi
´
ˆet H.
Vˆa
.
y l
`
’
oi tuyˆen b
´
ˆo c
’
ua nh`a s
’
an xu
´
ˆat l`a khˆong ¯d´ung s
’
u
.
thˆa
.
t.
6. KI
’
ˆ
EM D
¯
I
.
NH GI
’
A THI
´
ˆ
ET V
`
ˆ
E S
.
’
U B
`
˘
ANG NHAU GI
˜’
UA HAI
TRUNG B
`
INH
Gi
’
a s
’
’
u X v`a Y l`a hai ¯da
.
i l
’
u
’
o
.
ng ng
˜
ˆau nhiˆen ¯dˆo
.
c lˆa
.
p c´o c`ung phˆan ph
´
ˆoi chu
’
ˆan v
´
’
oi
E(X) v`a E(Y ) ch
’
ua bi
´
ˆet. Ta c
`
ˆan ki
’
ˆem ¯di
.
nh gi
’
a thi
´
ˆet
H : E(X) = E(Y ) (H : E(X) = E(Y ))
L
´
ˆay m˜au ng
˜
ˆau nhiˆen k´ıch th
’
u
´
’
oc n ¯d
´
ˆoi X v`a m
˜
ˆau ng
˜
ˆau nhiˆen k´ıch th
’
u
´
’
oc m ¯d
´
ˆoi v
´
’
oi
Y v`a x´et c´ac tr
’
u
`
’
ong h
’
o
.
p:
i) Tr
’
u
`
’
ong h
’
o
.
p bi
´
ˆet V ar(x) = σ
2
x
, V ar(y) = σ
2
y
T´ınh gi´a tri
.
quan s´at u
0
=
|x − y|

σ
2
x
n
+
σ
2
y
m
.
ii) Tr
’
u
`
’
ong h
’
o
.
p ch
’
ua bi
´
ˆet V ar(X), V ar(Y ).
T´ınh gi´a tri
.
quan s´at u
0
=
|x − y|

s

2
x
n
+
s

2
y
m
.
V
´
’
oi m
´
’
uc ´y ngh
˜
ia α cho tr
’
u
´
’
oc, x´ac ¯di
.
nh phˆan vi
.
chu
’
ˆan u
1−
α
2
.
Ta t`ım ¯d
’
u
’
o
.
c mi
`
ˆen b´ac b
’
o W
α
= { u : |u| > u
1−
α
2
}.
So s´anh u
0
v`a u
1−
α
2
* N
´
ˆeu u
0
> u
1−
α
2
th`ı b´ac b
’
o gi
’
a thi
´
ˆet H v`a th
`
’
ua nhˆa
.
n H.
7. Ki
’
ˆem ¯di
.
nh gi
’
a thi
´
ˆet v
`
ˆe s
.
’
u b
`
˘
ang nhau c
’
ua hai t
’
y l
.
ˆe 93
* N
´
ˆeu u
0
< u
1−
α
2
th`ı th
`
’
ua nhˆa
.
n H.
• V´ı du
.
7 Tro
.
ng l
’
u
’
o
.
ng s
’
an ph
’
ˆam do hai nh`a m´ay s
’
an xu
´
ˆat l`a c´ac ¯da
.
i l
’
u
’
o
.
ng ng
˜
ˆau
nhiˆen c´o phˆan ph
´
ˆoi chu
’
ˆan v`a c´o c`ung ¯dˆo
.
lˆe
.
ch tiˆeu chu
’
ˆan l`a σ = 1kg. V
´
’
oi m
´
’
uc ´y ngh
˜
ia
α = 0, 05, c´o th
’
ˆe xem tro
.
ng l
’
u
’
o
.
ng trung b`ınh c
’
ua s
’
an ph
’
ˆam do hai nh`a m´ay s
’
an xu
´
ˆat l`a
nh
’
u nhau hay khˆong? N
´
ˆeu cˆan th
’
’
u 25 s
’
an ph
’
ˆam c
’
ua nh`a m´ay A ta t´ınh ¯d
’
u
’
o
.
c x = 50kg,
cˆan 20 s
’
an ph
’
ˆam c
’
ua nh`a m´ay B th`ı t´ınh ¯d
’
u
’
o
.
c y = 50, 6kg.
Gi
’
ai
Go
.
i tro
.
ng l
’
u
’
o
.
ng c
’
ua nh`a m´ay A l`a X; tro
.
ng l
’
u
’
o
.
ng c
’
ua nh`a m´ay B l`a Y th`ı X, Y l`a
c´ac ¯da
.
i l
’
u
’
o
.
ng ng
˜
ˆau nhiˆen c´o phˆan ph
´
ˆoi chu
’
ˆan v
´
’
oi V ar(X) = V ar(Y ) = 1.
Ta ki
’
ˆem tra gi
’
a thi
´
ˆet H : E(X) = E(Y ); (E(X) = E(Y ))
V
´
’
oi m
´
’
uc ´y ngh
˜
ia α = 0, 05 th`ı u
1−
α
2
= 1, 96.
T´ınh u
0
=
|50−50,6|
√
1
25
+
1
20
= 2.
Ta th
´
ˆay u
0
> u
1−
α
2
nˆen b´ac b
’
o gi
’
a thi
´
ˆet H, t
´
’
uc l`a tro
.
ng l
’
u
’
o
.
ng trung b`ınh c
’
ua s
’
an
ph
’
ˆam s
’
an xu
´
ˆat
’
’
o hai nh`a m´ay l`a kh´ac nhau.
7. KI
’
ˆ
EM D
¯
I
.
NH GI
’
A THI
´
ˆ
ET V
`
ˆ
E S
.
’
U B
`
˘
ANG NHAU C
’
UA HAI
T
’
Y L
ˆ
E
.
Gi
’
a s
’
’
u p
1
, p
2
t
’
u
’
ong
´
’
ung l`a t
’
y lˆe
.
c´ac ph
`
ˆan t
’
’
u mang d
´
ˆau hiˆe
.
u n`ao ¯d´o c
’
ua t
’
ˆong th
’
ˆe
th
´
’
unh
´
ˆat, t
’
ˆong th
’
ˆe th
´
’
u hai. Ta c
`
ˆan ki
’
ˆem ¯di
.
nh gi
’
a thi
´
ˆet
H : p
1
= p
2
= p
0
(H : p
1
= p
2
)
i) Tr
’
u
`
’
ong h
’
o
.
p ch
’
ua bi
´
ˆet p
0
.
Cho
.
n th
´
ˆong kˆe U =
(P
∗
− p
1
) − (p
∗
− p
2
)

p
∗
(1 − p
∗
)(
1
n
1
+
1
n
2
)
.
v
´
’
oi p
∗
=
n
1
.f
n
1
+ n
2
.f
n
2
n
1
+ n
2
(
’
u
´
’
oc l
’
u
’
o
.
ng h
’
o
.
p l´y t
´
ˆoi ¯da c
’
ua p
0
)
trong ¯d´o
f
n
1
l`a t
’
y lˆe
.
ph
`
ˆan t
’
’
u c´o d
´
ˆau hiˆe
.
u c
’
ua m
˜
ˆau th
´
’
u nh
´
ˆat v
´
’
oi k´ıch th
’
u
´
’
oc n
1
.
f
n
2
l`a t
’
y lˆe
.
ph
`
ˆan t
’
’
u c´o d
´
ˆau hiˆe
.
u c
’
ua m
˜
ˆau th
´
’
u hai v
´
’
oi k´ıch th
’
u
´
’
oc n
2
.
V
´
’
oi n
1
, n
2
kh´a l
´
’
on th`ı U c´o phˆan ph
´
ˆoi chu
’
ˆan h´oa.
ii) Tr
’
u
`
’
ong h
’
o
.
p bi
´
ˆet p
0
.
Cho
.
n th
´
ˆong kˆe U =
f
n
1
− f
n
2

p
0
(1 − p
0
)(
1
n
1
+
1
n
2
)
94 Ch ’u ’ong 5. Ki
’
ˆem ¯di
.
nh gi
’
a thi
´
ˆet th
´
ˆong kˆe
* Qui t
´
˘
ac ki
’
ˆem ¯di
.
nh
L
´
ˆay hai m
˜
ˆau ng
˜
ˆau nhiˆen k´ıch th
’
u
´
’
oc n
1
, n
2
v`a t´ınh
u
0
=
|f
n
1
− f
n
2
|

p
∗
(1 − p
∗
)(
1
n
1
+
1
n
2
)
(p
∗
=
n
1
.f
n
1
+ n
2
.f
n
2
n
1
+ n
2
) n
´
ˆeu ch
’
ua bi
´
ˆet p
0
ho
˘
a
.
c
u
0
=
|f
n
1
− f
n
2

p
0
(1 − p
0
)(
1
n
1
+
1
n
2
)
n
´
ˆeu bi
´
ˆet p
0
.
V
´
’
oi m
´
’
uc ´y ngh
˜
ia α cho tr
’
u
´
’
oc, x´ac ¯di
.
nh phˆan vi
.
chu
’
ˆan u
1−
α
2
.
Ta t`ım ¯d
’
u
’
o
.
c mi
`
ˆen b´ac b
’
o W
α
= { u : |u|.u
1−
α
2
}.
So s´anh u
0
v`a u
1−
α
2
* N
´
ˆeu u
0
> u
1−
α
2
th`ı b´ac b
’
o gi
’
a thi
´
ˆet H.
* N
´
ˆeu u
0
< u
1−
α
2
th`ı th
`
’
ua nhˆa
.
n gi
’
a thi
´
ˆet H.
• V´ı du
.
8 Ki
’
ˆem tra c´ac s
’
an ph
’
ˆam ¯d
’
u
’
o
.
c cho
.
n ng
˜
ˆau nhiˆen
’
’
o hai nh`a m´ay s
’
an xu
´
ˆat ta
¯d
’
u
’
o
.
c c´ac s
´
ˆo liˆe
.
u sau:
Nh`a m´ay I S
´
ˆo s
’
an ph
’
ˆam ¯d
’
u
’
o
.
c ki
’
ˆem tra S
´
ˆo ph
´
ˆe ph
’
ˆam
I n
1
= 100 20
II n
2
= 120 36
V
´
’
oi m
´
’
uc ´y ngh
˜
ia α = 0, 01; c´o th
’
ˆe coi t
’
y lˆe
.
ph
´
ˆe ph
’
ˆam c
’
ua hai nh`a m´ay l`a nh
’
u nhau
khˆong?
Gi
’
ai
Go
.
i p
1
, p
2
t
’
u
’
ong
´
’
ung l`a t
’
y lˆe
.
ph
´
ˆe ph
’
ˆam c
’
ua nh`a m´ay I, II.
Ta ki
’
ˆem tra gi
’
a thi
´
ˆet H : p
1
= p
2
(H : p
1
= p
2
).
V
´
’
oi m
´
’
uc ´y ngh
˜
ia α = 0, 01 th`ı u
1−
α
2
= u
0,995
= 2, 58.
T
`
’
u c´ac s
´
ˆo liˆe
.
u ¯d˜a cho ta c´o
f
n
1
=
20
100
= 0, 2; f
n
2
=
36
120
= 0, 3
p
∗
=
100 × 0, 2 + 120 × 0, 3
100 + 120
= 0, 227 =⇒ 1 − p
∗
= 0, 773
Do ¯d´o u
0
=
|0, 2 − 0, 3|

0, 227 × 0, 773(
1
100
+
1
120
)
≈ 1, 763.
Ta th
´
ˆay u
0
< u
1−
α
2
nˆen ch
´
ˆap nhˆa
.
n gi
’
a thi
´
ˆet H, t
´
’
uc l`a t
’
y lˆe
.
ph
´
ˆe ph
’
ˆam c
’
ua hai nh`a
m´ay l`a nh
’
u nhau.
8. Ki
’
ˆem ¯di
.
nh gi
’
a thi
´
ˆet v
`
ˆe s
.
’
u b
`
˘
ang nhau gi
˜
’
ua hai ph
’
u
’
ong sai 95
8. KI
’
ˆ
EM D
¯
I
.
NH GI
’
A THI
´
ˆ
ET V
`
ˆ
E S
.
’
U B
`
˘
ANG NHAU GI
˜
’
UA HAI
PH
’
U
’
ONG SAI
Gi
’
a s
’
’
u X, Y l`a hai ¯da
.
i l
’
u
’
o
.
ng ng
˜
ˆau nhiˆen ¯dˆo
.
c lˆa
.
p c´o phˆan ph
´
ˆoi chu
’
ˆan v
´
’
oi c´ac tham s
´
ˆo
t
’
u
’
ong
´
’
ung σ
2
x
, σ
2
y
ch
’
ua bi
´
ˆet. Ta c
`
ˆan ki
’
ˆem ¯di
.
nh gi
’
a thi
´
ˆet
H : σ
2
x
= σ
2
y
(gi
’
a thi
´
ˆet ¯d
´
ˆoi H : σ
2
x
= σ
2
y
)
L
´
ˆay m
˜
ˆau ng
˜
ˆau nhiˆen W
X
= (X
1
, X
2
, . . . , X
n
), W
Y
= (Y
1
, Y
2
, . . . , Y
n
) ¯d
´
ˆoi v
´
’
oi X, Y .
Cho
.
n c´ac th
´
ˆong kˆe
S
2
x
=

n
i=1
(X
i
− X)
2
n − 1
S
2
y
=

m
i=1
(Y
j
− X)
2
m − 1
Ta th
´
ˆay
(n − 1)S
2
x
σ
2
x
v`a
(m − 1)S
2
y
σ
2
y
l`a c´ac ¯da
.
i l
’
u
’
o
.
ng ng
˜
ˆau nhiˆen ¯dˆo
.
c lˆa
.
p c´o phˆan ph
´
ˆoi
χ
2
v
´
’
oi n − 1 v`a m − 1 bˆa
.
c t
’
u
.
do. Do ¯d´o
S
2
x
/σ
2
x
S
2
y
/σ
2
y
c´o phˆan ph
´
ˆoi F v
´
’
oi c´ac tham s
´
ˆo n − 1
v`a m − 1.
Khi H ¯d´ung th`ı S
2
x
/S
2
y
∈ F
α/2,n−1,m−1
v`a c´o
P (F
1−α/2,n−1,m−1
< S
2
x
/S
2
y
< F
α/2,n−1,m−1
) = 1 − α
Ta t`ım ¯d
’
u
’
o
.
c
* Mi
`
ˆen b´ac b
’
o W
α
= (−∞, F
1−α/2,n−1,m−1
) ∪ (F
α/2,n−1,m−1
, +∞).
* Gi´a tri
.
quan s´at v =
S
2
x
S
2
y
Do ¯d´o
• N
´
ˆeu v ∈ W
α
th`ı b´ac b
’
o gi
’
a thi
´
ˆet H v`a ch
´
ˆap nhˆa
.
n H.
• N
´
ˆeu v /∈ W
α
th`ı ch
´
ˆap nhˆa
.
n gi
’
a= thi
´
ˆet H.
 Ch´u ´y Ki
’
ˆem ¯di
.
nh
’
’
o trˆen bi
.
’
anh h
’
u
’
’
ong b
’
’
oi gi´a tri
.
quan s´at v = S
2
x
/S
2
y
v`a x´ac su
´
ˆat
P (F
n−1,m−1
< v) trong ¯d´o F
n−1,m−1
l`a ¯da
.
i l
’
u
’
o
.
ng ng
˜
ˆau nhiˆen c´o phˆan ph
´
ˆoi F v
´
’
oi c´ac
tham s
´
ˆo n− 1, m− 1. N
´
ˆeu x´ac su
´
ˆat nh
’
o h
’
on
α
2
(x
’
ay ra khi S
2
x
nh
’
o h
’
on S
2
y
) ho
˘
a
.
c l
´
’
on h
’
on
1 − α/2 (x
’
ay ra khi S
2
x
l
´
’
on h
’
on S
2
y
) th`ı gi
’
a thi
´
ˆet bi
.
t
`
’
u ch
´
ˆoi.
N
´
ˆeu ¯d
˘
a
.
t
p − gi´a tri
.
= 2 min[P (F
n−1,m−1<v
), 1 − P (F
n−1,m−1
)]
th`ı gi
’
a thi
´
ˆet bi
.
t
`
’
u ch
´
ˆoi khi m
´
’
uc ´y ngh
˜
ia α t l
´
’
on h
’
on p−gi´a tri
.
.
• V´ı du
.
9 C´o hai c´ach cho
.
n ch
´
ˆat x´uc t´ac kh´ac nhau ¯d
’
ˆe k´ıch th´ıch mˆo
.
t ph
’
an
´
’
ung h´oa
ho
.
c. D
¯
’
ˆe ki
’
ˆem ¯di
.
nh ph
’
u
’
ong sai s
’
an sinh ra c´o gi
´
ˆong nhau hay khˆong ng
’
u
`
’
oi ta l
´
ˆay m
˜
ˆau
g
`
ˆom 10 nh´om d`ung cho ch
´
ˆat x´uc t´ac th
´
’
u nh
´
ˆat v`a 12 nh´om d`ung cho ch
´
ˆat x´uc t´ac th
´
’
u hai.
96 Ch ’u ’ong 5. Ki
’
ˆem ¯di
.
nh gi
’
a thi
´
ˆet th
´
ˆong kˆe
D
˜
’
u liˆe
.
u cho k
´
ˆet qu
’
a S
2
1
= 0, 14 v`a S
2
2
= 0, 28. V
´
’
oi m
´
’
uc ´y ngh
˜
ia 5%, h˜ay ki
’
ˆem ¯di
.
nh gi
’
a
thi
´
ˆet trˆen.
Gi
’
ai
Ta c
`
ˆan ki
’
ˆem ¯di
.
nh gi
’
a thi
´
ˆet H : σ
2
1
= σ
2
2
.
Ta c´o v =
S
2
1
S
2
2
=
0,14
0,28
= 0, 5 v`a P (F
9,11<0,5
) = 0, 1539.
Do ¯d´o p−gi´a tri
.
= 2 min(0, 1539; 0, 8461) = 0, 3074.
Ta th
´
ˆay α = 0, 05 < p−gi´a tri
.
nˆen gi
’
a thi
´
ˆet v
`
ˆe s
’
u
.
b
`
˘
ang nhau c
’
ua hai ph
’
u
’
ong sai
¯d
’
u
’
o
.
c ch
´
ˆap nhˆa
.
n.
9. B
`
AI T
ˆ
A
.
P
1. D
¯
ˆo
.
b
`
ˆen c
’
ua mˆo
.
t loa
.
i dˆay th´ep s
’
an xu
´
ˆat theo cˆong nghˆe
.
c˜u l`a 150. Sau khi c
’
ai ti
´
ˆen
k˜y thuˆa
.
t ng
’
u
`
’
oi ta l
´
ˆay m
˜
ˆau g
`
ˆom 100 s
’
o
.
i dˆay th´ep ¯d
’
ˆe th
’
’
u ¯dˆo
.
b
`
ˆen th`ı th
´
ˆay ¯dˆo
.
b
`
ˆen
trung b`ınh l`a 185 v`a s = 25. V
´
’
oi m
´
’
uc ´y ngh
˜
ia α = 0, 05, h
’
oi cˆong nghˆe
.
m
´
’
oi c´o t
´
ˆot
h
’
on cˆong nghˆe
.
c˜u hay khˆong?
2. D
¯
ˆo
.
d`ay c
’
ua mˆo
.
t chi ti
´
ˆet m
´
ˆay do mˆo
.
t m´ay s
’
an xu
´
ˆat l`a mˆo
.
t ¯da
.
i l
’
u
’
o
.
ng ng
˜
ˆau nhiˆen
phˆan ph
´
ˆoi theo qui luˆa
.
t chu
’
ˆan v
´
’
oi ¯dˆo
.
d`ay trung b`ıng 1, 25mm. Nghi ng
`
’
o m´ay hoa
.
t
¯dˆo
.
ng khˆong b`ınh th
’
u
`
’
ong ng
’
u
`
’
oi ta ki
’
ˆem tra 10 chi ti
´
ˆet m´ay th`ı th
´
ˆay ¯dˆo
.
d`ai trung
b`ınh l`a 1, 325 v
´
’
oi ¯dˆo
.
lˆe
.
ch tiˆeu chu
’
ˆan 0, 075mm. V
´
’
oi m
´
’
uc ´y ngh
˜
ia α = 0, 01, h˜ay
k
´
ˆet luˆa
.
n v
`
ˆe ¯di
`
ˆeu nghi ng
`
’
o n´oi trˆen?
3. Tro
.
ng l
’
u
’
o
.
ng c
’
ua mˆo
.
t loa
.
i s
’
an ph
’
ˆam do mˆo
.
t nh`a m´ay s
’
an xu
´
ˆat l`a ¯da
.
i l
’
u
’
o
.
ng ng
˜
ˆau
nhiˆen phˆan ph
´
ˆoi theo qui luˆa
.
t chu
’
ˆan v
´
’
oi tro
.
ng l
’
u
’
o
.
ng trung b`ınh l`a 500 gr. Nghi
ng
`
’
o tro
.
ng l
’
u
’
o
.
ng c
’
ua loa
.
i s
’
an ph
’
ˆam n`ay c´o xu h
’
u
´
’
ong gi
’
am s´ut, ng
’
u
`
’
oi ta cˆan th
’
’
u 25
s
’
an ph
’
ˆam v`a thu ¯d
’
u
’
o
.
c k
´
ˆet qu
’
a cho
’
’
o b
’
ang sau:
Tro
.
ng l
’
u
’
o
.
ng (gr) 480 485 490 495 500 510
S
´
ˆo s
’
an ph
’
ˆam 2 3 8 5 3 4
V
´
’
oi m
´
’
uc ´y ngh
˜
ia α = 0, 05, h˜ay k
´
ˆet luˆa
.
n v
`
ˆe ¯di
`
ˆeu nghi ng
`
’
o n´oi trˆen?
4. N
˘
ang su
´
ˆat l´ua trung b`ınh trong vu
.
tr
’
u
´
’
oc l`a 4,5 t
´
ˆan/ha. Vu
.
l´ua n
˘
am nay ng
’
u
`
’
oi
ta ´ap du
.
ng mˆo
.
t biˆe
.
n ph´ap k˜y thuˆa
.
t m
´
’
oi cho to`an bˆo
.
diˆe
.
n t´ıch tr
`
ˆong l´ua
’
’
o trong
v`ung. Theo d˜oi n
˘
ang su
´
ˆat l´ua
’
’
o 100 hecta ta c´o b
’
ang s
´
ˆo liˆe
.
u sau:
9. B`ai t
.
ˆap 97
N
˘
ang su
´
ˆat (ta
.
/ha) Diˆe
.
n t´ıch (ha)
30 − 35 7
35 − 40 12
40 − 45 18
45 − 50 27
50 − 55 20
55 − 60 8
60 − 65 5
65 − 70 3
H˜ay cho k
´
ˆet luˆa
.
n v
`
ˆe biˆe
.
n ph´ap k˜y thuˆa
.
t m
´
’
oi n`ay?
5. Tu
’
ˆoi tho
.
trung b`ınh c
’
ua mˆo
.
t m
˜
ˆau g
`
ˆom 100 b´ong ¯d`en ¯d
’
u
’
o
.
c s
’
an xu
´
ˆat
’
’
o mˆo
.
t nh`a
m´ay l`a 1570 gi
`
’
o v
´
’
oi ¯dˆo
.
lˆe
.
ch tiˆeu chu
’
ˆan 120 gi
`
’
o. Go
.
i µ l`a tu
’
ˆoi tho
.
trung b`ınh c
’
ua
t
´
ˆat c
’
a b´ong ¯d`en nh`a m´ay s
’
an xu
´
ˆat ra. V
´
’
oi m
´
’
uc ´y ngh
˜
ia α = 0, 05, h˜ay ki
’
ˆem tra
gi
’
a thi
´
ˆet H
0
: µ = 1600 gi
`
’
o v
´
’
oi gi
’
a thi
´
ˆet ¯d
´
ˆoi H
1
: µ < 1600 gi
`
’
o.
6. Mˆo
.
t h˜ang d
’
u
’
o
.
c ph
’
ˆam s
’
an xu
´
ˆat mˆo
.
t loa
.
i thu
´
ˆoc tri
.
di
.
´
’
ung th
’
u
.
c ph
’
ˆam tuyˆen b
´
ˆo r
`
˘
ang
thu
´
ˆoc c´o t´ac du
.
ng gi
’
am di
.
´
’
ung trong 8 gi
`
’
o ¯d
´
ˆoi v
´
’
oi 90% ng
’
u
`
’
oi d`ung. Ki
’
ˆem tra 200
ng
’
u
`
’
oi bi
.
di
.
´
’
ung d`ung th`ı th
´
ˆay thu
´
ˆoc c´o t´ac du
.
ng ¯d
´
ˆoi v
´
’
oi 160 ng
’
u
`
’
oi . V
´
’
oi m
´
’
uc ´y
ngh
˜
ia α = 0, 01, ki
’
ˆem tra xem l
`
’
oi tuyˆen b
´
ˆo trˆen c´o ¯d´ung khˆong?
7. T
’
y lˆe
.
ph
´
ˆe ph
’
ˆam c
’
ua mˆo
.
t nh`a m´ay tr
’
u
´
’
oc ¯dˆay l`a 5%. N
˘
am nay nh`a m´ay ´ap du
.
ng
mˆo
.
t biˆe
.
n ph´ap k˜y thuˆa
.
t m
´
’
oi. D
¯
’
ˆe xem biˆe
.
n ph´ap k˜y thuˆa
.
t m
´
’
oi c´o t´ac du
.
ng l`am
gi
’
am t
’
y lˆe
.
ph
´
ˆe ph
’
ˆam c
’
ua nh`a m´ay hay khˆong, ng
’
u
`
’
oi ta l
´
ˆay mˆo
.
t m
˜
ˆau g
`
ˆom 800 s
’
an
ph
’
ˆam ¯d
’
ˆe ki
’
ˆem tra v`a th
´
ˆay c´o 24 ph
´
ˆe ph
’
ˆam trong m
˜
ˆau n`ay.
a) V
´
’
oi m
´
’
uc ´y ngh
˜
ia α = 0, 01, h˜ay cho k
´
ˆet luˆa
.
n v
`
ˆe biˆe
.
n ph´ap k˜y thuˆa
.
t m
´
’
oi ¯d´o?
b) N
´
ˆeu nh`a m´ay b´ao c´ao t
’
y lˆe
.
ph
´
ˆe ph
’
ˆam sau khi ´ap du
.
ng biˆe
.
n ph´ap k˜y thuˆa
.
t m
´
’
oi
¯d˜a gi
’
am xu
´
ˆong 2% (v
’
os i m
´
’
uc ´y ngh
˜
ia α = 0, 05) th`ı c´o ch
´
ˆap nhˆa
.
n ¯d
’
u
’
o
.
c khˆong?
8. Gi´am ¯d
´
ˆoc mˆo
.
t nh`a m´ay tuyˆen b
´
ˆo 90% m´ay m´oc c
’
ua nh`a m´ay ¯da
.
t tiˆeu chu
’
ˆan k˜y
thuˆa
.
t qu
´
ˆoc t
´
ˆe. Ng
’
u
`
’
oi ta ti
´
ˆen h`anh ki
’
ˆem tra 200 m´ay th`ı th
´
ˆay c´o 168 m´ay ¯da
.
t tiˆeu
chu
’
ˆan k˜y thuˆa
.
t qu
´
ˆoc t
´
ˆe. V
´
’
oi m
´
’
uc ´y ngh
˜
ia α = 0, 05, h˜ay k
´
ˆet luˆa
.
n v
`
ˆe l
`
’
oi tuyˆen b
´
ˆo
trˆen?
9. N
´
ˆeu m´ay m´oc l`am viˆe
.
c b`ınh th
’
u
`
’
ong th`ı k´ıch th
’
u
´
’
oc c
’
ua mˆo
.
t loa
.
i s
’
an ph
’
ˆam l`a ¯da
.
i
l
’
u
’
o
.
ng ng
˜
ˆau nhiˆen phˆan ph
´
ˆoi theo qui luˆa
.
t chu
’
ˆan v
´
’
oi V ar(X) = 0, 25. Nghi ng
`
’
o
m´ay l`am viˆe
.
c khˆong b`ınh th
’
u
`
’
ong, ng
’
u
`
’
oi ta ti
´
ˆen h`anh ¯do th
’
’
u 28 s
’
an ph
’
ˆam v`a thu
¯d
’
u
’
o
.
c k
´
ˆet qu
’
a cho
’
’
o b
’
ang sau:
K´ıch th
’
u
´
’
oc (cm) 19,0 19,5 19,8 20,4 20,6
S
´
ˆo s
’
an ph
’
ˆam 2 4 5 12 5
V
´
’
oi m
´
’
uc ´y ngh
˜
ia α = 0, 02, h˜ay k
´
ˆet luˆa
.
n v
`
ˆe ¯di
`
ˆeu nghi ng
`
’
o n´oi trˆen?

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