Giải bài tập chương 8 xác suất thống kê trong sách bài tập

µ σ
2
= 64
H
0
: µ = 52
H
1
: µ = 52
W
α
=

U =
(
X − 52)
√
n
σ
; |U| > u
α/2

n = 25; x = 55, 4
⇒ U
qs
=
(55, 4 −52)
√
25
8
= 2, 125

α = 0, 05 ⇒ u
α/2
= u
0,025
= 1, 96 ⇒ W
α
= (−∞; −1, 96) ∪ (1, 96; +∞)
U
qs
∈ W
α
H
0
H
1
µ σ
2
H
0
: µ = 2, 5; H
1
: µ < 2, 5
W
α
=

T =
(
X − 2, 5)
√

n
S
; T < −t
(n−1)
α

n = 100;
x = 2, 455; s = 0, 3 → T
qs
=
(2, 455 −2, 5)
√
100
0, 3
= −1, 5
α = 0, 05 → t
(n−1)
α
≈ u
α
= u
0,05
= 1, 645 → W
α
= (−∞; −1, 645)
T
qs
/∈ W
α
H

0
σ
2
= 45
2
H
0
: µ = 400; H
1
: µ = 400
W
α
=

U =
(
X − 400)
√
n
σ
; |U| > u
α/2

n = 25; x = 395 → U
qs
=
(395 − 400)
√
25
45

= −0, 55 6
α = 0, 05 → u
α/2
= u
0,025
= 1, 96 → W
α
= (−∞; −1, 96) ∪ (1, 96; +∞)
U
qs
/∈ W
α
H
0
→
β = P

U < u
α/2
−
|µ
0
− µ
1
|
√
n
σ

= P


U < 1, 96 −
|400 − 3 90|.5
45

= P[U < 0, 85 ] = Φ
0
(0, 85) + 0, 5 = 0, 3023 + 0, 5 = 0, 8033
µ σ
2
= 20
2
H
0
: µ = 20; H
1
: µ = 20
W
α
=

U =
(
X − 20 )
√
n
σ
; |U| > u
α/2


n = 100;
x =
19.10 + 20.60 + 21.20 + 22.5 + 23. 5
100
= 20, 35
U
qs
=
(20, 35 − 20).10
2
= 1, 75
α = 0, 05 → u
α/2
= u
0,025
= 1, 96 → W
α
= (−∞; −1, 96) ∪ (1, 96; +∞)
U
qs
/∈ W
α
µ σ
2
= 36
2
H
0
: µ = 453; H
1

: µ < 453
W
α
=

U =
(
X − 45 3)
√
n
σ
; U < −u
α

n = 81;
x = 448 → U
qs
=
(448 − 453).9
36
= −1, 25
α = 0, 05 → u
α
= u
0,05
= 1, 645 → W
α
= (−∞; −1, 645)
U
qs

/∈ W
α
µ σ
2
H
0
: µ = 14; H
1
: µ < 14
W
α
=

T =
(
X − 14 )
√
n
S
; T < −t
(n−1)
α

n = 25 ;
x = 12, 5; s = 2, 5 → T
qs
=
(12, 5 − 14)
√
25

2, 5
= −3
α = 0, 05 → t
(n−1)
α
= t
(24)
0,05
1, 711 → W
α
= (−∞; −1, 711)
T
qs
∈ W
α
H
0
µ σ
2
H
0
: µ = 44; H
1
: µ < 44
W
α
=

T =
(

X − 44 )
√
n
S
; T < −t
(n−1)
α

n = 25;
x = 41, 6; s = 3, 2 → T
qs
=
(41, 6 − 44)
√
25
3, 2
= −3, 75
α = 0, 01 → t
(n−1)
α
= t
(24)
0,01
2, 492 → W
α
= (−∞; −2, 492)
T
qs
∈ W
α

H
0
µ σ
2
H
0
: µ = 47; H
1
: µ < 47
W
α
=

T =
(
X − 47 )
√
n
S
; T < −t
(n−1)
α

n = 25;
x = 45, 5; s = 4 → T
qs
=
(45, 5 − 47)
√
25

4
= −1, 87 5
α = 0, 05 → t
(n−1)
α
= t
(24)
0,05
= 1, 711 → W
α
= (−∞; −1, 711)
T
qs
∈ W
α
H
0
µ σ
2
H
0
: µ = 50; H
1
: µ < 50
W
α
=

T =
(

X − 50 )
√
n
S
; T < −t
(n−1)
α

n = 30 ; x = 49, 533; s = 0, 552 → T
qs
=
(49, 533 − 50)
√
30
0.552
= −4, 62 9
α = 0, 05 → t
(n−1)
α
= t
(29)
0,05
= 1, 699 → W
α
= (−∞; −1, 699)
T
qs
∈ W
α
H

0
= P

T < t
(n−1)
α
−
|µ
0
− µ
1
|
√
n
s

= P

T < 1, 699 −
|50 − 48|.
√
30
0, 552

= P[T < −18, 14] = P [T > 18, 14] = 0, 001
µ σ
2
H
0
: µ = 14; H

1
: µ = 14
W
α
=

T =
(
X − 14 )
√
n
S
; |T | > t
(n−1)
α/2

n = 25 ;
x = 15; s =
√
5 = 2, 2 36 → T
qs
=
(15 − 14)
√
25
2, 236
= −2, 23 6
α = 0, 05 → t
(n−1)
α/2

= t
(24)
0,025
= 2, 064 → W
α
= (−∞; −2, 064) ∪ (2, 064; +∞ )
T
qs
∈ W
α
H
0
σ
2
= 1, 2
2
H
0
: µ = 16; H
1
: µ = 16
W
α
=

U =
(
X − 16 )
√
n

σ
; |U| > u
α/2

n = 25;
x = 16, 5 → U
qs
=
(16, 5 − 16)
√
25
1, 2
= 2, 083
α = 0, 05 → u
α/2
= u
0,025
= 1, 96 → W
α
= (−∞; −1, 96) ∪ (1, 96; +∞)
U
qs
∈ W
α
H
0
µ
1
= 15, 5
→ β = P


U < 1, 96−
|16 − 15, 5|.5
1, 2

= P[U < −0, 123] = P[ U > 0, 123] = 0, 4522
µ
1
= 16, 6
→ β = P

U < 1, 96 −
(16, 6 − 16).5
1, 2

= P[U < −0, 54] = P [U > 0, 54] = 0, 2946
α = 0, 05; β = 0, 02, ∆ = 1
→ n 
σ
2
(u
α/2
+ u
β
)
2
∆
2
=
1, 2

2
(1, 96 + 2, 05)
2
1
2
= 23, 16
µ σ
2
H
0
: µ = 32000; H
1
: µ = 32000
W
α
=

T =
(
X − 32000)
√
n
S
; |T | < t
(n−1)
α/2

n = 16 ;
x = 34625; s = 3200 → T
qs

=
(34625 − 32000)
√
16
3200
= 3, 28125
α = 0, 05 → t
(n−1)
α/2
= t
(15)
0,025
= 2, 131 → W
α
= (−∞; −2, 131) ∪ (2, 131; +∞ )
T
qs
∈ W
α
H
0
P
value
= 2P[ T > |T
qs
|] = 2P [T > 3, 28125] = 2.0, 0 05 = 0, 01 < 0, 05 = α
H
0
µ σ
2

H
0
: µ = 20, 2; H
1
: µ > 20, 2
W
α
=

T =
(
X − 20, 2)
√
n
S
; T > t
(n−1)
α

n = 12 ;
x = 20, 625; s = 0, 191 → T
qs
=
(20, 625 − 20, 2)
√
12
0, 191
= 7, 708
α = 0, 05 → t
(n−1)

α/2
= t
(11)
0,05
= 1, 796 → W
α
= (1, 796; +∞)
T
qs
∈ W
α
H
0
µ
0
= 210; x = 218; s
2
=
1
n − 1

(x
i
−
x)
2
=
1
24
· 24 00 = 100 → s = 10

→ T
qs
=
(218 − 210).
√
25
10
= 4
P
value
= P[T > T
qs
] = P [T > 4] < 0, 001
H
0
µ σ
2
H
0
: µ = 1000; H
1
: µ = 1000
W
α
=

T =
(
X − 10 00)
√

n
S
; T > t
(n−1)
α

n = 64 ;
x = 0; s = 100 → T
qs
=
(990 − 1000)
√
64
100
= −0, 8
α = 0, 05 → t
(n−1)
α/2
= t
(63)
0,025
≈ u
0,025
= 1, 96 → W
α
= (−∞; −1, 96) ∪ (1, 96; +∞)
T
qs
/∈ W
α

H
0
P
value
= 2P [T > |T
qs
|] = 2.P [T > 0, 8] ≈ 2P[U > 0, 8] = 2.0, 2119 = 0, 4238 >
α
H
0
β = P

T < 1, 96 −
|1050−1000|.8
100

= P[T < −2, 04] ≈ P[U > 2, 04] = 0, 0207
1 −β = 1−P

T < 1, 96 −
|980−1000|.8
100

= 1−P [T < 0 , 36] = 1 −0, 3594 = 0, 6406
α = β = 0, 05, ∆ = 30
n 
s
2
(t
(n−1)

α/2
+ t
(n−1)
β
)
2
∆
2
=
100
2
(1, 96 + 1, 6 45)
2
30
2
= 144, 4
X
1
, X
2
X
1
∼ N(µ
1
, σ
2
1
); X
1
∼ N(µ

2
, σ
2
2
)
µ σ
2
H
0
: µ
1
= µ
2
; H
1
: µ
1
> µ
2
n
1
= n
2
= 36 > 30
W
α
=

U =
(

X
1
− X
2
)

S
2
1
n
1
+
S
2
2
n
2
; U > u
α

x
1
= 12, 5; x
2
= 12, 2; s
1
= 1, 2; s
2
= 1, 4;
U

qs
=
(12, 5 − 12, 2)

1,2
2
36
+
1,4
2
36
= 0, 976;
α = 0, 01 → u
α
= u
0,01
= 2, 33 → W
α
= (2, 33; +∞)
U
qs
/∈ W
α
H
0
X
1
, X
2
X

1
∼ N(µ
1
, σ
2
1
); X
1
∼ N(µ
2
, σ
2
2
)
µ σ
2
H
0
: µ
1
= µ
2
; H
1
: µ
1
< µ
2
n
1

= 64 > 30; n
2
= 68 > 30
W
α
=

U =
(
X
1
− X
2
)

S
2
1
n
1
+
S
2
2
n
2
; U < −u
α

x

1
= 373, 2; x
2
= 76, 6; s
1
= 10, 9s
2
= 11, 2;
U
qs
=
(73, 2 − 76, 6)

10,9
2
64
+
11,2
2
68
= −1, 76 7;
α = 0, 05 → u
α
= u
0,05
= 1, 645 → W
α
= (−∞; −1, 645)
U
qs

∈ W
α
H
0
|µ
1
− µ
2
| = 2
β = P

U < 1, 645 −
2

10,9
2
64
+
11,2
2
68

= P[U < 0, 61]
= 1 − P[U > 0, 61] = 1 − 0, 27 09 = 0, 7291
X
1
, X
2
X
1

∼ N(µ
1
, σ
2
1
); X
1
∼ N(µ
2
, σ
2
2
)
µ σ
2
H
0
: µ
1
= µ
2
; H
1
: µ
1
< µ
2
n
1
= 8000 > 30; n

2
= 2000 > 30
W
α
=

U =
(
X
1
− X
2
)

S
2
1
n
1
+
S
2
2
n
2
; U < −u
α

x
1

= 3, 0; x
2
= 3, 2; s
1
= 0, 9; s
2
= 0, 4;
U
qs
=
(3, 0 − 3, 2)

0,9
2
8000
+
0,4
2
2000
= −14, 856;
α = 0, 05 → u
α
= u
0,05
= 1, 645 → W
α
= (−∞; −1, 645)
U
qs
∈ W

α
H
0
X
1
, X
2
X
1
∼ N(µ
1
, σ
2
1
); X
1
∼ N(µ
2
, σ
2
2
)
µ σ
2
H
0
: µ
1
= µ
2

; H
1
: µ
1
< µ
2
n
1
= 100 > 30; n
2
= 150 > 30
W
α
=

U =
(
X
1
− X
2
)

S
2
1
n
1
+
S

2
2
n
2
; U < −u
α

x
1
= 1, 1; x
2
= 1, 2; s
1
= 0, 2; s
2
= 0, 3;
U
qs
=
(1, 1 − 1, 2)

0,2
2
100
+
0,3
2
150
= −3, 16 2;
α = 0, 05 → u

α
= u
0,05
= 1, 645 → W
α
= (−∞; −1, 645)
U
qs
∈ W
α
H
0
X
1
, X
2
X
1
∼ N(µ
1
, σ
2
1
); X
1
∼ N(µ
2
, σ
2
2

)
µ σ
2
H
0
: µ
1
= µ
2
; H
1
: µ
1
= µ
2
n
1
= 1000 > 30; n
2
= 500 > 30
W
α
=

U =
(
X
1
− X
2

)

S
2
1
n
1
+
S
2
2
n
2
; |U| > u
α/2

x
1
= 70; x
2
= 72; s
1
= 10; s
2
= 20;
U
qs
=
(70 − 72)


10
2
1000
+
20
2
500
= 2, 108
α = 0, 05 → u
α/2
= u
0,025
= 1, 96 → W
α
= (−∞; −1, 96) ∪ (1, 96; +∞)
U
qs
∈ W
α
H
0
x
1
x
2
X
1
, X
2
X

1
∼ N(µ
1
, σ
2
1
); X
1
∼ N(µ
2
, σ
2
2
)
µ σ
2
H
0
: µ
1
= µ
2
; H
1
: µ
1
> µ
2
n
1

= 100 > 30; n
2
= 50 > 30
W
α
=

U =
(
X
1
− X
2
)

S
2
1
n
1
+
S
2
2
n
2
; U > u
α

x

1
= 100; x
2
= 95; s
1
= 10; s
2
= 9;
U
qs
=
(100 − 95)

10
2
100
+
9
2
50
= 3, 089
α = 0, 05 → u
α
= u
0,05
= 1, 645 → W
α
= (1, 645; +∞)
U
qs

∈ W
α
H
0
X
1
, X
2
X
1
∼ N(µ
1
, σ
2
1
); X
1
∼ N(µ
2
, σ
2
2
)
µ σ
2
H
0
: µ
1
= µ

2
; H
1
: µ
1
= µ
2
n
1
= 38 > 30; n
2
= 40 > 30
W
α
=

U =
(
X
1
− X
2
)

S
2
1
n
1
+

S
2
2
n
2
; |U| > u
α/2

x
1
= 89, 7; x
2
= 94, 5; s
1
= 12, 2; s
2
= 13, 05;
U
qs
=
(89, 7 − 94 , 5)

12,2
2
38
+
13,05
2
40
= −1, 679

α = 0, 05 → u
α/2
= u
0,025
= 1, 96 → W
α
= (−∞; −1, 96) ∪ (1, 96; +∞)
U
qs
/∈ W
α
H
0
X
1
, X
2
X
1
∼ N(µ
1
, σ
2
1
); X
1
∼ N(µ
2
, σ
2

2
)
µ σ
2
H
0
: µ
1
= µ
2
; H
1
: µ
1
> µ
2
W
α
=

T =
(X
1
− X
2
)

S
2
1

n
1
+
S
2
2
n
2
; T > t
(k)
α

x
1
= 5, 16; x
2
= 4, 61; s
1
= 0, 267; s
2
= 0, 179;
T
qs
=
(5, 16 − 4, 61)

0,267
2
10
+

0,179
2
10
= 5, 402
C =
0,072
10
0,072
10
+
0,032
10
= 0, 69; k =
(10 − 1)(10 − 1)
9.0, 69
2
+ 9.0, 31
2
= 15, 73
k = 16; α = 0, 01 → t
(k)
α
= t
(16)
0,01
= 2, 583 → W
α
= (2, 583; +∞)
U
qs

∈ W
α
H
0
σ
2
1
= σ
2
2
W
α
=

T =
(
X
1
− X
2
)
S
p

1
n
1
+
1
n

2
; T > t
(n−1+n
2
−2)
α

S
p
=

(n
1
− 1) S
2
1
+ (n
2
− 1)S
2
2
n
1
+ n
2
− 2
=

9.0, 072 + 9.0, 032
18

= 0, 228
→ T
qs
=
(5, 16 − 4, 61)
0, 228.

1
10
+
1
10
= 5, 393
t
(n
1
+n
2
−2)
α
= t
(18)
0,01
= 2, 552 → W
α
= (2, 552; +∞)
T
qs
∈ W
α

H
0
X
1
, X
2
X
1
∼ N(µ
1
, σ
2
1
); X
1
∼ N(µ
2
, σ
2
2
)
µ D = X
1
− X
2
H
0
: µ
1
− µ

2
= 0; H
1
: µ
1
− µ
2
< 0
W
α
=

T =
D
√
n
S
D
; T < −t
(n−1)
α

d = −20; s
D
= 17, 89 → T
qs
=
−20.
√
6

17, 89
= −2, 73 8
α = 0, 01 → t
(n−1)
α
= t
(5)
0,01
= 3, 365 → W
α
= (−∞; −3, 365)
U
qs
/∈ W
α
H
0
X
1
, X
2
X
1
∼ N(µ
1
, σ
2
1
= 0, 517
2

); X − 2 ∼ N(µ
2
, σ
2
2
= 0, 485
2
)
µ
H
0
: µ
1
= µ
2
; H
1
: µ
1
= µ
2
W
α
=

U =
X
1
+ X
2


σ
2
1
n
1
+
σ
2
2
n
2
; |U| > u
α/2

x
1
= 1, 317; x
2
= 1, 512; U
qs
=
1, 317 − 1, 5 12

0,517
2
230
+
0,485
2

302
= −1, 75 1;
α = 0, 01 → u
α/2
= u
0,005
= 2, 57 → W
α
= (−∞; −2, 57) ∪ (2, 57; +∞)
U
qs
/∈ W
α
|µ
1
− µ
2
| = 0, 3
β = P

U < 2, 57 −
0, 3

0,517
2
230
+
0,485
2
302


= P[U < −0, 12 ] = P [U > 0, 12] = 0, 4522
X
1
, X
2
X
1
∼ N(µ
1
, σ
2
1
); X
1
∼ N(µ
2
, σ
2
2
)
µ σ
2
H
0
: µ
1
= µ
2
; H

1
: µ
1
= µ
2
W
α
=

T =
(
X
1
− X
2
)

S
2
1
n
1
+
S
2
2
n
2
; |T | > t
(k)

α/2

x
1
= 4, 7667; x
2
= 5, 2125; s
2
1
= 0, 4547; s
2
2
= 0, 2984;
T
qs
=
(4, 7667 − 5, 2125)

0,4547
6
+
0,2984
8
= −1, 32 56
C =
0,4547
6
0,4547
6
+

0,2984
8
= 0, 67; k =
(6 − 1)(8 − 1)
6.0, 67
2
+ 7.0, 33
2
≈ 10
k = 10; α = 0, 05 → t
(k)
α/2
= t
(10)
0,025
= 2, 228 → W
α
= (−∞; −2.228) ∪ (2 , 228; +∞)
T
qs
/∈ W
α
H
0
X
1
, X
2
X
1

∼ N(µ
1
, σ
2
1
); X
1
∼ N(µ
2
, σ
2
2
)
µ σ
2
H
0
: µ
1
= µ
2
; H
1
: µ
1
= µ
2
W
α
=


T =
(
X
1
− X
2
)

S
2
1
n
1
+
S
2
2
n
2
; |T | > t
(k)
α/2

x
1
= 6, 6; x
2
= 7, 2; s
2

1
= 1, 3; s
2
2
= 0, 7;
T
qs
=
(6, 6 − 7, 2)

1,3
5
+
0,7
5
= −0, 95
C =
1,3
5
01,3
5
+
0,7
5
= 0, 65; k =
(5 − 1)(5 − 1)
4.0, 65
2
+ 4. 0, 35
2

= 7, 34
k = 8; α = 0, 05 → t
(k)
α/2
= t
(10)
0,025
= 2, 306 → W
α
= (−∞; −2.306) ∪ (2, 306; +∞)
T
qs
/∈ W
α
H
0
X
1
, X
2
X
1
∼ N(µ
1
, σ
2
1
); X
1
∼ N(µ

2
, σ
2
2
)
µ σ
2
H
0
: µ
1
= µ
2
; H
1
: µ
1
> µ
2
W
α
=

T =
(
X
1
− X
2
)


S
2
1
n
1
+
S
2
2
n
2
; T > t
(k)
α

x
1
= 185, 1; x
2
= 177; s
2
1
= 286, 7667; s
2
2
= 53, 1111;
T
qs
=

(185, 1 − 177)

286,7667
5
+
53,1111
5
= 1, 389
C =
286,7667
5
286,7667
5
+
53,1111
5
= 0, 844; k =
(10 − 1)(10 − 1)
9.0, 844
2
+ 9. 0, 156
2
= 12, 22
k = 13; α = 0, 1 → t
(k)
α
= t
(13)
0,1
= 1, 35 → W

α
= (1, 35; +∞)
T
qs
∈ W
α
H
0
H
0
: p = 0, 03; H
1
: p > 0, 03
W
α
=

U =
(f − 0, 03)
√
n

0, 03(1 − 0 , 03)
; U > U
α

→ f = 14/40 0 = 0, 035
→ U
qs
=

(0, 035 − 0 , 03)
√
400

0, 03(1 − 0, 03 )
= 0, 568
α = 0, 05 → U
α
= U
0,05
= 1, 65 → W
α
= (1, 65; +∞)
U
qs
/∈ W
α
H
0
H
0
: p = 0, 03; H
1
: p > 0, 03
W
α
=

U =
(f − 0, 03)

√
n

0, 03(1 − 0 , 03)
; U > U
α

→ f = 12/17 0 = 0, 0706
→ U
qs
=
(0, 0706 − 0, 03)
√
170

0, 03(1 − 0, 03 )
= 3, 103
α = 0, 01 → U
α
= U
0,01
= 2, 33 → W
α
= (2, 33; +∞)
U
qs
∈ W
α
H
0

β = P

U < U
α
−
|p
1
− p
0
|
√
n

p
0
(1 − p
0
)

= p

U < 2, 33 −
|0, 05 − 0, 03|
√
170
√
0, 03.0 , 97

= P[U < 0, 8] = 1 −P [U > 0, 8] = 1 −0, 2 119 = 0, 7881
H

0
: p = 0, 05; H
1
: p > 0, 05
U
qs
=
(f − p
0
)
√
n

p
0
(1 − p
0
)
=
(0, 04 − 0, 05)
√
100
√
0, 05.0 , 95
= −0, 46
P
value
= P(U > U
qs
) = P (U > −0, 46) = 1 −P (U > 0, 46) = 1 −0, 3228 = 0, 677 2

p
1
= 7%, α = 0, 05, β = 0, 0 2
n 
f(1 − f)
|p
1
− p
0
|
2
[(u
α
+ u
β
]
2
=
0, 04.0 , 96
|0, 07 − 0, 05|
2
(1, 65 + 2, 0 5)
2
= 1314, 24
H
0
: p = 0, 05; H
1
: p > 0, 05
W

α
=

U =
(f − 0, 05)
√
n
√
0, 05.0 , 95
; U > u
α

→ f =
24
300
= 0, 08
⇒ U
qs
=
(0, 08 − 0, 05)
√
300
√
0, 05.0 , 95
= 2, 384
α = 0, 05 → U
α
= u
0,05
= 1, 65 → W

α
= (1, 65; +∞) ⇒ U
qs
∈ W
α
H
0
H
0
: p = 0, 06; H
1
: p < 0, 06
W
α
=

U =
(f − 0, 06)
√
n
√
0, 06.0 , 94
; U < −u
α

→ f =
5
100
= 0, 05
⇒ U

qs
=
(0, 05 − 0, 06)
√
100
√
0, 06.0 , 94
= −0, 42 2
α = 0, 05 → U
α
= u
0,05
= 1, 65 → W
α
= (−∞; −1, 65) ⇒ U
qs
/∈ W
α
H
0
H
0
: p = 0, 85; H
1
: p > 0, 85
W
α
=

U =

(f − 0, 85)
√
n
√
0, 85.0 , 15
; U < −u
α

→ f =
810
900
= 0, 9
⇒ U
qs
=
(0, 9 − 0, 85)
√
900
√
0, 85.0 , 15
= 4, 2
α = 0, 05 → U
α
= u
0,05
= 1, 65 → W
α
= (1, 65; +∞) ⇒ U
qs
∈ W

α
H
0
H
0
: p = 0, 5; H
1
: p = 0, 5
W
α
=

U =
(f − 0, 5)
√
n
√
0, 5.0, 5
; |U| > u
α/2

⇒ U
qs
=
(0, 7 − 0, 5)
√
100
√
0, 5.0, 5
= 4

α = 0, 01 → U
α/2
= u
0,005
= 2, 57 → W
α
= (−∞; −2, 57) ∪ (2, 57; +∞)
⇒ U
qs
∈ W
α
H
0
p
A
, p
B
H
0
: p
A
= p
B
; H
1
: p
A
< p
B
W

α
=

U =
f
A
− f
B

f(1 − f)

1
n
A
+
1
n
B

; U < −u
α

f
A
=
30
200
= 0, 15; f
B
=

65
350
= 0, 186 →
f =
30 + 6 5
200 + 350
= 0, 173
⇒ U
qs
=
(0, 15 − 0, 186)

0, 173. 0, 827

1
200
+
1
350

= −1, 07 4
α = 0, 05 → u
α
= u
0,05
= 1, 65 → W
α
= (−∞; −1, 65)
U
qs

/∈ W
α
H
0
p
1
, p
2
H
0
: p
1
= p
2
; H
1
: p
1
= p
2
f
1
=
30
100
= 0, 3; f
2
=
40
150

= 0, 267 →
f =
30 + 4 0
100 + 15 0
= 0, 28
⇒ U
qs
=
(0, 3 − 0, 267)

0, 28.0 , 72

1
100
+
1
150

= 0, 569
P
value
= P(U > |U
qs
|) = P (U > 0, 569) = 0, 2843
p
1
, p
2
H
0

: p
1
= p
2
; H
1
: p
1
< p
2
W
α
=

U =
f
1
− f
2

f(1 − f)

1
n
1
+
1
n
2


; U < −u
α

f
1
=
5
50
= 0, 1; f
2
=
7
40
= 0, 175 →
f =
5 + 7
50 + 40
= 0, 133
⇒ U
qs
=
(0, 1 − 0, 175)

0, 133. 0, 867

1
50
+
1
40


= −1, 04 1
α = 0, 01 → u
α
= u
0,01
= 2, 33 → W
α
= (−∞; −2, 33)
U
qs
/∈ W
α
H
0
p
A
, p
B
H
0
: p
A
= p
B
; H
1
: p
A
< p

B
W
α
=

U =
f
A
− f
B

f(1 − f)

1
n
A
+
1
n
B

; U < −u
α

f
A
=
175
1900
= 0, 092; f

B
=
325
2600
= 0, 125 →
f =
175 + 32 5
1900 + 2 600
= 0, 111
⇒ U
qs
=
(0, 092 − 0, 12 5)

0, 111. 0, 889

1
1900
+
1
2600

= −3, 48
α = 0, 05 → u
α
= u
0,05
= 1, 65 → W
α
= (−∞; −1, 65)

U
qs
∈ W
α
H
0
∆ = 0, 02
β =P

U < u
α
−
∆

f(1 − f)

1
n
A
+
1
n
B


=P

U < 1, 65 −
0, 02


0, 111. 0, 889

1
1900
+
1
2600


=P (U < −0, 47) = P (U > 0, 47) = 0, 3192
p
1
, p
2
H
0
: p
1
= p
2
; H
1
: p
1
= p
2
W
α
=


U =
f
1
− f
2

f(1 − f)

1
n
1
+
1
n
2

; |U| > u
α/2

f
1
=
54
1800
= 0, 03; f
2
=
30
1200
= 0, 025 →

f =
54 + 3 0
1800 + 1 200
= 0, 028
⇒ U
qs
=
(0, 03 − 0, 025)

0, 028. 0, 972

1
1800
+
1
1200

= 0, 813
α = 0, 05 → u
α/2
= u
0,025
= 1, 96 → W
α
= (−∞; −1, 96) ∪ (1, 96; +∞)
U
qs
/∈ W
α
H

0
p
1
, p
2
H
0
: p
1
= p
2
; H
1
: p
1
= p
2
W
α
=

U =
f
1
− f
2

f(1 − f)

1

n
1
+
1
n
2

; |U| > u
α/2

f
1
=
20
200
= 0, 1; f
2
=
120
800
= 0, 15 →
f =
20 + 120
200 + 800
= 0, 14
⇒ U
qs
=
(0, 1 − 0, 15)


0, 14.0 , 86

1
200
+
1
800

= −1, 82 3
α = 0, 05 → u
α/2
= u
0,025
= 1, 96 → W
α
= (−∞; −1, 96) ∪ (1, 96; +∞)
U
qs
/∈ W
α
H
0
p
′
1
p
2
H
0
: p

1
= p
2
; H
1
: p
1
= p
2
W
α
=

U =
f
1
− f
2

f(1 − f)

1
n
1
+
1
n
2

; |U| > u

α/2

f
1
=
140
1000
= 0, 14; f
2
=
260
2000
= 0, 13 →
f =
140 + 260
1000 + 2 000
= 0, 133
⇒ U
qs
=
(0, 14 − 0, 13)

0, 133. 0, 867

1
1000
+
1
2000


= 0, 76
α = 0, 05 → u
α/2
= u
0,025
= 1, 96 → W
α
= (−∞; −1, 96) ∪ (1, 96; +∞)
U
qs
/∈ W
α
H
0
n = 15, s
2
= 144, α = 0, 01
H
0
: σ
2
= 138; H
1
: σ
2
> 138
W
α
=


χ
2
=
(n − 1)S
2
138
; χ
2
> χ
2(n−1)
α

χ
2(n−1)
α
= χ
2(14)
0,01
= 29, 14 → W
α
= (29, 14; +∞)
χ
2
qs
=
(n − 1)s
2
138
=
14.144

138
= 14, 61 → χ
2
qs
/∈ W
α
H
0
→ X ∼ N(µ, σ
2
)
σ
2
µ
H
0
: σ
2
= 1000
2
; H
1
: σ
2
> 1000
2
W
α
=


χ
2
=
(n − 1)S
2
1000
2
; χ
2
> χ
2(n−1)
α

n = 10 ; s = 1150; α = 0, 05
χ
2(n−1)
α
= χ
2(9)
0,05
= 16, 92 → W
α
= (16, 92; +∞)
χ
2
qs
=
(n − 1)s
2
1000

2
=
9.1150
2
1000
2
= 11, 9 → χ
2
qs
/∈ W
α
H
0
→ X ∼ N(µ, σ
2
)
σ
2
µ
H
0
: σ
2
= 0, 05
2
; H
1
: σ
2
> 0, 05

2
W
α
=

χ
2
=
(n − 1)S
2
0, 05
2
; χ
2
> χ
2(n−1)
α

χ
2(n−1)
α
= χ
2(15)
0,01
= 30, 58 → W
α
= (30, 58; +∞)
n = 16, s
2
= 0, 0775, α = 0, 01

χ
2
qs
=
(n − 1)s
2
0, 05
2
=
15.0, 0775z
2
0, 05
2
= 23, 25 → χ
2
qs
/∈ W
α
H
0
→ X ∼ N(µ, σ
2
)
σ
2
σ
2
H
0
: σ

2
= 10; H
1
: σ
2
> 10
W
α
=

χ
2
=
(n − 1)S
2
σ
2
0
; χ
2
> χ
2(n−1)
α

n = 12 ; s
2
= 11, 41
χ
2
qs

=
11.11 , 41
10
= 12, 55
α = 0, 05 → χ
2(n−1)
α
= χ
2(11)
0,05
= 19, 68 → W
α
= (19, 68; +∞)
χ
2
qs
/∈ W
α
H
0
X ∼ N(µ, σ
2
)
σ
2
µ
H
0
: σ
2

= 1
2
; H
1
: σ
2
> 1
2
W
α
=

χ
2
=
(n − 1)S
2
1
2
; χ
2
> χ
2(n−1)
α

n = 30 ; s = 1, 1; α = 0, 01
χ
2(n−1)
α
= χ

2(29)
0,01
= 49, 59 → W
α
= (49, 59; +∞)
χ
2
qs
=
(n − 1)s
2
1
2
=
15.1, 1
2
1
2
= 35, 09 → χ
2
qs
/∈ W
α
H
0
x
A
= 137, 29; s
2
A

= 3, 905
x
B
= 139; s
B
= 5
X
A
, X
B
→ X
A
∼ N(µ
A
, σ
2
A
); X
B
∼
N(µ
B
, σ
2
B
) σ
2
A
; σ
2

B
H
0
: σ
2
A
= σ
2
B
; H
1
: σ
2
A
= σ
2
B
W
α
=

F =
S
2
B
S
2
A
; F < f
(n

B
−1,n
A
−1)
1−α/2
F > f
(n
B
−1,n
A
−1)
α/2

F
qs
=
s
2
B
s
2
A
=
5
3, 905
= 1, 28
f
(n
B
−1,n

A
−1)
1−α/2
= f
(6,6)
0,975
=
1
f
(6,6)
0,025
=
1
5, 82
= 0, 172
f
(n
B
−1,n
A
−1)
α/2
= f
(6,6)
0,025
= 5, 82
W
α
= (0; 0, 172) ∪ (5, 82; +∞) → F
qs

/∈ W
α
X
A
, X
B
→ X
A
∼ N(µ
A
, σ
2
A
); X
B
∼
N(µ
B
, σ
2
B
)
σ
2
A
; σ
2
B
H
0

: σ
2
A
= σ
2
B
; H
1
: σ
2
A
= σ
2
B
W
α
=

F =
S
2
B
S
2
A
; F < f
(n
B
−1,n
A

−1)
1−α/2
F > f
(n
B
−1,n
A
−1)
α/2

F
qs
=
s
2
B
s
2
A
=
17, 2
14, 5
= 1, 186
f
(n
B
−1,n
A
−1)
1−α/2

= f
(19,24)
0,975
=
1
f
(24,19)
0,025
=
1
2, 45
= 0, 41
f
(n
B
−1,n
A
−1)
α/2
= f
(19,24)
0,025
= 2, 44
W
α
= (0; 0, 41) ∪ (2, 44; +∞) → F
qs
/∈ W
α
s

2
A
=
1
n
A
− 1


12
i=1
x
2
i1
− n
a
.
x
2
A

=
1
11
(49 − 12.1, 5
2
) = 2
s
2
B

=
1
n
B
− 1


i=1
15x
2
i2
− n
B
.
x
2
B

=
1
14
(158 − 15 .2
2
) = 7
X
A
, X
B
→ X
A

∼ N(µ
A
, σ
2
A
); X
B
∼ N(µ
B
, σ
2
B
)
σ
2
A
; σ
2
B
H
0
: σ
2
A
= σ
2
B
; H
1
: σ

2
A
= σ
2
B
F
qs
=
s
2
B
s
2
A
=
7
2
= 3, 5
P
value
= P(F > |F
qs
|) = P (F > 3, 5) > 0, 025
H
0
n
1
= 4, n
2
= 7; s

2
1
= 85, 576; s
2
2
= 13, 78
X
1
, X
2
→ X
1
∼
N(µ
1
, σ
2
1
); X
2
∼ N(µ
2
, σ
2
2
)
σ
2
1
, σ

2
2
H
0
: σ
2
1
= σ
2
2
; H
1
: σ
2
1
> σ
2
2
W
α
=

F =
S
2
1
S
2
2
; F > f

(n
1
−1;n
2
−1)
α

F
qs
=
s
2
1
s
2
2
=
85, 576
13, 78
= 6, 21
α = 0, 05 → f
(n
1
−1;n
2
−1)
α
= f
(3,6)
0,05

= 4, 76 → W
α
= (4, 76; +∞)
F
qs
∈ W
α
H
0
X
A
, X
B
X
A
∼ N(µ
A
, σ
2
A
); X
B
∼
N(µ
B
, σ
2
B
)
H

0
: σ
2
A
= σ
2
B
; H
1
: σ
2
B
.σ
2
A
( s
2
B
> s
2
A
)
W
α
=

F =
S
2
B

S
2
A
; F > f
(n
B
−1,n
a
−1)
α

n
A
= 10; n
B
= 15; α = 0, 05 → f
(n
B
−1,n
A
−1)
α
= f
(14,9)
0,05
= 3, 01 → W
α
= (3, 01; +∞)
F
qs

=
s
2
B
s
2
A
=
16
1,44
= 11, 11 ⇒ F
qs
∈ W
α
H
0
H
0
H
1
W
α
=

χ
2
= n

h


i=1
k

j=1
n
2
ij
n
i
n
j
− 1

; χ
2
> χ
2(h−1)(k−1)
α

h = 3, k = 3, n = 356
χ
2
qs
=356

20
2
140.104
+
53

2
140.128
+
67
2
140.124
+
52
2
131.104
+
47
2
131.128
+
+
32
2
131.124
+
32
2
85.104
+
28
2
85.128
+
25
2

85.124
− 1

=356[1, 0833 − 1] = 29, 647 9
α = 0, 1 → χ
2(h−1)(k−1)
α
= χ
2(3−1)(3−1)
0,1
= χ
2(4)
0,1
= 7, 779 → W
α
= (7, 779; +∞)
→ χ
2
qs
∈ W
α
H
0
H
0
H
1
W
α
=


χ
2
= n

h

i=1
k

j=1
n
2
ij
n
i
n
j
− 1

; χ
2
> χ
2(h−1)(k−1)
α

h = 2, k = 2, n = 571
χ
2
qs

=571

123
2
276.268
+
153
2
276.303
+
145
2
295.268
+
150
2
295.303
− 1

=571[1, 00211 − 1 ] = 1, 205
α = 0, 01 → χ
2(h−1)(k−1)
α
= χ
2(2−1)(2−1)
0,01
= χ
2(1)
0,01
= 6, 635 → W

α
= (6, 635; +∞)
→ χ
2
qs
/∈ W
α
H
0
H
0
H
1
W
α
=

χ
2
= n

h

i=1
k

j=1
n
2
ij

n
i
n
j
− 1

; χ
2
> χ
2(h−1)(k−1)
α

h = 3, k = 3, n = 542
χ
2
qs
=542

180
2
281.250
+
58
2
281.188
+
43
2
281.104
+

34
2
144.250
+
76
2
144.188
+
+
34
2
144.104
+
36
2
117.250
+
54
2
117.188
+
27
2
144.104
− 1

=542[1, 1455 − 1] = 78, 867 9
α = 0, 05 → χ
2(h−1)(k−1)
α

= χ
2(3−1)(3−1)
0,05
= χ
2(4)
0,05
= 9, 488 → W
α
= (9, 488; +∞)
→ χ
2
qs
∈ W
α
H
0
H
0
H
1
W
α
=

χ
2
= n

h


i=1
k

j=1
n
2
ij
n
i
n
j
− 1

; χ
2
> χ
2(h−1)(k−1)
α

h = 3, k = 2, n = 148
χ
2
qs
=148

20
2
49.58
+
25

2
58.58
+
13
2
31.58
+
29
2
49.90
+
43
2
68.90
+
18
2
31.90
− 1

=148[1, 0022 − 1] = 0, 3203
α = 0, 05 → χ
2(h−1)(k−1)
α
= χ
2(3−1)(2−1)
0,05
= χ
2(2)
0,05

= 5, 991 → W
α
= (5, 991; +∞)
→ χ
2
qs
/∈ W
α
H
0
H
0
H
1
W
α
=

χ
2
= n

h

i=1
k

j=1
n
2

ij
n
i
n
j
− 1

; χ
2
> χ
2(h−1)(k−1)
α

h = 4, k = 2, n = 400
χ
2
qs
=400

28
2
72.124
+
42
2
120.124
+
30
2
108.124

+
24
2
100.124
+
44
2
72.276
+
78
2
120.276
+
78
2
108.276
+
76
2
100.276
− 1

=400[1, 0145 − 1] = 5, 81