Giải bài tập chương 7 xác suất thống kê trong sách bài tập


V (X) = 500
W = (X
1
, X
2
, , X
n
) X
1
, X
2
, , X
100
E(X
i
) = E(X) = 1065, V (X
i
) = V (X) = 500
2
, ∀i = 1, 100
X =
1
100
(X
1
+ X
2
+ + X
100
)

E(X) =
1
100
(E(X
1
) + E(X
2
) + + E(X
100
)) =
1
100
.100.1065 = 1065
V (
X) =
1
100
2
(V (X
1
) + V (X
1
) + + V (X
100
)) =
1
100
2
.100.500
2

= 2500
W = (X
1
, X
2
, , X
n
) X
i
V (X) = V

X
1
+ + X
n
n

=
1
n
2
[V (X
1
) + + V (X
n
)]
=
1
n
2

nV (X) =
V (X)
n
=
1000000
n
Se(X)  25 ⇔

V (X)  25 ⇔

1000000
n
 25
⇔ n 
1000000
25
2
= 1600
Se(f) =

V (f) =

p(1 − p)
√
n
=
√
0, 5.0, 5
√
120

=
0, 5
10, 9445
= 0, 0456
W = (X
1
, X
2
, X
3
, X
4
, X
5
)
X
x =
x
1
+ + x
5
5
=
10 + 12 + 16 + 18 + 19
5
= 15
f =
3
50
= 0, 06

ˆ
θ
∧θ
BS(
ˆ
θ) = 1/n, BS(∧θ) = 0, 01
BS(
ˆ
θ) < BS(∧θ) ⇔
1
n
< 0, 01 ⇔ n > 100
V (X) = σ
2
X
d
V (X
d
) =
1
4n[f(x
d
)]
2
=
1
4.500[f(x
d
)]
2

=
1
2000[f(x
d
)]
2
X
′
V (X
′
) =
V (X)
320
=
σ
2
320
EF =
V (
X
′
)
V (X
d
)
=
σ
2
320
1

2000[f (x
d
)]
2
=
2000
320
[f(x
d
)]
2
σ
2
= 6, 25[f(x
d
)]
2
σ
2
= 6, 25[f(µ)]
2
σ
2
= 6, 25
1
σ
2
.2π
σ
2

=
6, 25
2π
< 1
X ∼ N(µ, σ
2
)
W = (X
1
, X
2
, , X
n
)
X
d
V (X
d
) =
1
4n[f(x
d
)]
2
=
2π
4
V (
X)
X V (X)

X
d
X
EF =
V (
X)
V (X
d
)
=
V (
X)
2π
4
V (
X)
=
4
2π
≈ 0, 64
E(X) = 8.
1
4
+ 10.
1
4
+ 11.
1
2
= 10

E(Y ) = 4.
1
2
+ 6.
1
2
= 5
E(Z) = 32
1
8
+ 40
1
8
+ 44
1
4
+ 48
1
8
+ 60
1
8
+ 66
1
8
= 50
X ∼ A(p)
W
1
= (X

11
, X
12
, X
13
) n
1
= 3
W
2
= (X
21
, X
22
, X
23
, X
24
) n
2
= 4
f
1
, f
2
E(f
1
) = E(f
2
) = p

V (f
1
) =
p(1 − p)
n
1
=
p(1 − p)
3
; V (f
2
) =
p(1 − p)
n
2
=
p(1 − p)
4
V (f
2
) < V (f
1
) f
2
f
1
f
2
f
1

EF =
V (f
1
)
V (f
2
)
=
p(1−p)
3
p(1−p)
4
=
4
3
≈ 1, 333
W
1
W
2
θ = αf
1
+ (1 − α)f
2
V (θ) = V (αf
1
+ (1 − α)f
2
) = α
2

V (f
1
) + (1 − α)
2
V (f
2
)
= α
2
p(1 − p)
3
+ (1 − α)
2
p(1 − p)
4
= (7α
2
− 6α + 3)
p(1 − p)
12
g(α) = 7α
2
− 6α + 3 g(α) g(α)
α = 6/14 = 3/7 V (θ)
3
7
f
1
+
4

7
f
2
σ(V ) = 3σ(U) := 3. σ
(i) W
1
=
1
2
U +
1
2
V ⇒ E(W
1
) =
1
2
E(U) +
1
2
E(V ) = X
(ii) W
2
=
3
4
U +
1
4
V ⇒ E(W

2
) =
3
4
E(U) +
1
4
E(V ) = X
(iii) W
3
= 1.U + 0.V ⇒ E(W
3
) = E(U) = X
W
1
, W
2
, W
3
W
1
=
1
2
U +
1
2
V
V (W
1

) =
1
4
V (U) +
1
4
V (V ) =
σ
2
4
+
9σ
2
4
=
10
4
σ
2
W
2
=
3
4
U +
1
4
V
V (W
2

) =
9
16
V (U) +
1
16
V (V ) =
9σ
2
16
+
9σ
2
16
=
18
16
σ
2
W
3
= 1.U + 0.V
V (W
3
) = V (U) = σ
2
V (W
3
) < V (W
2

) < V (W
1
) W
3
µ
X
x =
17 + 28 + 92 + 41
4
= 44, 5
f =
8
20
= 0, 4
X ∼ P (λ)
f(x
i
, λ) = e
−λ
λ
x
i
x
i
!
, i =
1, 3
a. L(x
1
, x

2
, x
3
, λ) = f(x
1
, λ)f(x
2
, λ)f(x
3
, λ) = e
−3λ
λ
x
1
+x
2
+x
3
x
1
!x
2
!x
3
!
x
1
= 15, x
2
= 8, x

3
= 13
L(λ) = e
−3λ
λ
15+8+13
15!8!13!
= e
−3λ
λ
36
15!8!13!
b. L(λ = 5) = e
−15
5
36
15!8!13!
≈ 1, 356.10
−8
L(λ = 10) = e
−30
10
36
15!8!13!
≈ 2, 8501.10
−4
L(λ = 12) = e
−36
12
36

15!8!13!
= 5, 0075.10
−4
L(λ = 15) = e
−45
15
36
15!8!13!
≈ 1, 9043.10
−4
L(λ = 20) = e
−60
20
36
15!8!13!
≈ 1, 8328.10
−6
L(λ = 25) = e
−75
25
36
15!8!13!
≈ 1, 728.10
−9
λ = 12 =
x
1
+x
2
+x

3
3
L(x
1
, x
2
, , x
n
, λ) = e
−nλ
λ
n

i=1
x
i
x
1
!x
2
! x
n
!
ln L(x
1
, x
2
, , x
n
, λ) = −nλ + (ln λ)

n

i=1
x
i
− ln(x
1
!x
2
! x
n
!)
∂ ln L(x
1
, x
2
, , x
n
, λ)
∂λ
= −n +
1
λ
n

i=1
x
i
∂ ln L(x
1

, x
2
, , x
n
, λ)
∂λ
= 0 ⇔ λ =
1
n
n

i=1
x
i
=
x
∂
2
ln L(x
1
, x
2
, , x
n
, λ)
∂λ
2
= −
1
λ

2
n

i=1
x
i
⇒
∂
2
ln L(x
1
, x
2
, , x
n
, λ)
∂λ
2




λ=
x
= −
n
x
x
2
< 0

x λ
X ∼ E(λ) W = (X
1
, , X
5
)
f(x
i
, λ) =



λe
−λx
i
x
i
 0 i = 1, 5
0 x
i
< 0
x
i
< 0
x
i
 0, ∀i
L(x
1
, x

2
, x
3
, x
4
, x
5
, λ) =
5

i=1
f(x
i
, λ) = λ
5
e
−λ
5

i=1
x
i
x
1
= 1, 2; x
2
= 7, 5; x
3
= 1, 8; x
4

= 3, 7; x
5
= 0, 8
L(λ) = λ
5
e
−15λ
• λ = 0, 1 ⇒ L(λ = 0, 1) = 0, 1
5
e
−15.0,1
= 0, 1
5
e
−1,5
≈ 2, 2313.10
−6
• λ = 0, 2 ⇒ L(λ = 0, 2) = 0, 2
5
.e
−15.0,2
= 0, 2
5
e
−3
≈ 15, 9319.10
−6
• λ = 0, 3 ⇒ L(λ = 0, 3) = 0, 3
5
e

−15.0,3
= 0, 3
5
e
−4,5
≈ 26, 9947.10
−6
• λ = 0, 4 ⇒ L(λ = 0, 4) = 0, 4
5
e
−15.0,4
= 0, 4
5
.e
−6
≈ 25, 3824.10
−6
• λ = 0, 5 ⇒ L(λ = 0, 5) = 0, 5
5
.e
−15.0,5
= 0, 5
5
e
−7,5
≈ 17, 2839.10
−6
L(λ = 0, 3)
L(x
1

, , x
n
, λ) =
n

i=1
f(x
i
, λ) = λ
n
e
−λ
n

i=1
x
i
ln L(x
1
, , x
n
, λ) = n ln λ − λ
n

i=1
x
i
∂ ln L(x
1
, , x

n
, λ)
∂λ
=
n
λ
−
n

i=1
x
i
=
n
λ
− n
x
∂ ln L(x
1
, , x
n
, λ)
∂λ
= 0 ⇔
n
λ
− n
x = 0 ⇔ λ =
1
x

∂
2
ln L(x
1
, , x
n
, λ)
∂λ
2
= −
n
λ
2
< 0, ∀λ
λ 1/x
L(λ = 0, 3)
L(λ) λ =
1
x
=
5
1,2+7,5+1,8+3,7+0,8
=
1
3
L

1
3


=

1
3

5
e
−15.(1/3)
=

1
3

5
e
−5
≈ 27, 7282.10
−6
X ∼ B(n, p)
W = (X
1
, , X
m
)
X
i
X
i
∼
A(p), ∀i = 1, n

P (X
i
= x
i
) = f(x
i
, p) = p
x
i
(1 − p)
1−x
i
; x
i
= 0, 1.
X =
m

i=1
X
i
, f =
X
m
f = 0, 6 ⇒ x = f.m = 0, 6. 5 = 3
L(x
1
, , x
5
, p) =

5

i=1
f(x
i
, p) =
5

i=1
p
x
i
(1 − p)
1−x
i
= p

5
i=1
x
i
(1 − p)
5−

5
i=1
x
i
= p
x

(1 − p)
5−x
= p
3
(1 − p)
5−3
= p
3
(1 − p)
2
L(p) = p
3
(1 − p)
2
L(0) = 0
3
.1
2
= 0
L(0, 1) = 0, 1
3
.0, 9
2
= 81.10
−5
L(0, 2) = 0, 2
3
.0, 8
2
= 512.10

−5
L(0, 3) = 0, 3
3
.0, 7
2
= 1323.10
−5
L(0, 4) = 0, 4
3
.0, 6
2
= 2304.10
−5
L(0, 5) = 0, 5
3
.0, 5
2
= 3125.10
−5
L(0, 6) = 0, 6
3
.0, 4
2
= 3456.10
−5
L(0, 7) = 0, 7
3
.0, 3
2
= 3087.10

−5
L(0, 8) = 0, 8
3
.0, 2
2
= 2048.10
−5
L(0, 9) = 0, 9
3
.0, 1
2
= 729.10
−5
L(1) = 1
3
.0
2
= 0
L(x
1
, , x
m
, p) =
m

i=1
f(x
i
, p) =
m


i=1
p
x
i
(1 − p)
1−x
i
= p

m
i=1
x
i
(1 − p)
m−

m
i=1
x
i
= p
x
(1 − p)
m−x
ln L(x
1
, , x
m
, p) = ln


p
x
(1 − p)
m−x

= x ln p + (m − x) ln(1 − p)
∂ ln L(x
1
, , x
m
, p)
∂p
= x
1
p
+ (m − x)
−1
1 − p
=
x − mp
p(1 − p)
∂ ln L(x
1
, , x
m
, p)
∂p
= 0 ⇔
x − mp)

p(1 − p)
= 0 ⇔ p =
x
m
= f
∂
2
ln L(x
1
, , x
m
, p)
∂p
2
=
2xp − x − mp
2
[p(1 − p)]
2
∂
2
ln L(x
1
, , x
m
, p)
∂p
2





p=x/m
=
2
x
2
m
− x −
x
2
m
[p(1 − p)]
2
=
x(
x
m
− 1)
[p(1 − p)]
2
< 0
x
m
= f < 1
X ∼ N(µ, σ
2
= 2
2
)

µ
σ
2

X −
σ
√
n
u
α/2
;
X +
σ
√
n
u
α/2

x = 8, 5; σ = 2; n = 600;
α = 0, 05 ⇒ u
α/2
= u
0,025
= 1, 96

8, 5 −
2
√
600
· 1, 96; 8, 5 +

2
√
600
· 1, 96

= (8, 34; 8, 66)
X ∼ N(µ, σ
2
)
µ
σ
2
n  30

X −
S
√
n
· t
(n−1)
α/2
;
X +
S
√
n
· t
(n−1)
α/2


n = 30; α = 0, 05 ⇒ t
(n−1)
α/2
= t
(29)
0,025
= 2, 045
x =
304
30
= 10, 1333
s
2
=
1
29
(3082, 14 − (10, 1333)
2
.30) = 0, 0554 ⇒ s =

0, 0554 = 0, 235

10.1333 −
0, 235
√
30
· 2, 045; 10.1333 +
0, 235
√
30

· 2, 045

= (10, 045; 10, 221)
µ σ
2

X −
S
√
n
· t
(n−1)
α/2
;
X +
S
√
n
· t
(n−1)
α/2

n = 25; α = 0, 05 ⇒ t
(n−1)
α/2
= t
(24)
0,025
= 2, 064
x =

538
25
= 21, 52
s
2
=
1
24
(11716 −21, 52
2
.25) = 5, 76
s =

5, 76 = 2, 4

21, 52 −
2, 4
√
25
· 2, 064; 21, 52 +
2, 4
√
25
· 2, 064

= (20, 53; 22, 51)
µ σ
2

X −

S
√
n
· t
(n−1)
α/2
;
X +
S
√
n
· t
(n−1)
α/2

n = 25; α = 0, 05 ⇒ t
(n−1)
α/2
= t
(24)
0,025
= 2, 064
x =
962, 5
25
= 38, 5
s
2
=
1

24
(37706, 25 −38, 5
2
.25) = 27, 0833
s =

27, 0833 = 5, 204

38, 5 −
5, 204
√
25
· 2, 064; 38, 5 +
5, 204
√
25
· 2, 064

= (36, 352; 40, 648)
µ σ
2

X −
S
√
n
· t
(n−1)
α/2
;

X +
S
√
n
· t
(n−1)
α/2

n = 25; α = 0, 05 ⇒ t
(24)
0,025
= 2, 064;
x = 40; s = 5

40 −
5
√
25
· 2, 064; 40 +
5
√
25
· 2, 064

= (37, 936; 42, 064)
µ σ
2

X −
S

√
n
· t
(n−1)
α/2
;
X +
S
√
n
· t
(n−1)
α/2

n = 25; α = 0, 05 ⇒ t
(n−1)
α/2
= t
(4)
0,025
= 2, 776
x =
1
5
5

i=1
x
i
= 2, 018

s
2
=
1
4

5

i=1
x
2
i
− n.
x
2

= 2.10
−5
⇒ s = 0, 0045

2, 018 −
0, 0045
√
5
· 2, 776; 2, 018 −
0, 0045
√
5
· 2, 776


= (2, 0124; 2, 0236)
µ σ
2

X −
S
√
n
· t
(n−1)
α/2
;
X +
S
√
n
· t
(n−1)
α/2

n = 100; α = 0, 05 ⇒ t
(n−1)
α/2
= t
(99)
0,025
≈ u
0,025
= 1, 96
x =

1
n
k

i=1
n
i
x
i
=
1
100
· 1175, 5 = 11, 755
s
2
=
1
99
(13820, 05 −100.11, 755
2
) = 0, 0207 ⇒ s =

0, 0207 = 0, 144

11, 755 −
0, 144
√
100
· 1, 96; 11, 755 +
0, 144

√
100
· 1, 96

= (11, 727; 11, 783)
µ σ
2

X −
S
√
n
· t
(n−1)
α/2
;
X +
S
√
n
· t
(n−1)
α/2

n = 100; α = 0, 05 ⇒ t
(n−1)
α/2
= t
(99)
0,025

≈ u
0,025
= 1, 96
x =
1
100
· 9072 = 90, 72
s
2
=
1
99
(824732 −100.90, 72
2
) = 17, 3754 ⇒ s = 4, 168

90, 72 −
4, 168
10
· 1, 96; 90, 72 +
4, 168
10
· 1, 96

= (89, 903; 91, 537)
µ σ
2
x
= 3
2


X −
σ
√
n
u
α/2
;
X +
σ
√
n
u
α/2

n = 36, α = 0, 05 ⇒ u
α/2
= u
0,025
= 1, 96

x −
3
6
· 1, 96;
x +
3
6
· 1, 96


= (
x − 0, 98; x + 0, 98)
α = 0, 01 ⇒ u
0,005
= 2, 58
I
o
= 0, 6cm
n 

4σ
2
I
2
o
u
2
α/2

=

4.3
2
0, 6
2
2, 58
2

= [665, 64]
µ σ

2
x
= 1, 2
2
ε =
σ
√
n
u
α/2
ε = 0, 3 1 − α = 0, 95
n 

σ
2
ε
2
u
2
α/2

=

1, 2
2
0, 3
2
· 1, 96
2


= [61, 5]
X ∼ N(µ, σ
2
= 400)
ε
0
 15
⇒ α = 0, 05 ⇒ u
α/2
= u
0,025
= 1, 96
n 

σ
2
ε
2
0
u
2
α/2

=

400
15
2
· 1, 96
2


= [6, 83]
X ∼ N(µ, σ
2
) µ, σ
2
µ σ
2
n < 30 µ

x −
S
√
n
t
(n−1)
α
, +∞

(1 − α) µ x −
S
√
n
t
(n−1)
α
x =
265
25
= 10, 6

s
2
=
1
24
(2913 − 25.10, 6
2
) =
104
24
⇒ s = 2, 082
1 − α = 0, 95 ⇒ t
(n−1)
α
= t
(24)
0,05
= 1, 711
µ
µ > 10, 6 −
2, 082
5
· 1, 711 = 9, 89
x =
450
25
= 18
s
2
=

1
24
(8244 − 25.18
2
) = 6 ⇒ s =
√
6 = 2, 45
1 − α = 0, 95 ⇒ t
(n−1)
α
= t
(24)
0,05
= 1, 711
µ (1 − α)

− ∞;
x +
S
√
n
t
(n−1)
α

µ
x +
s
√
n

t
(n−1)
α
= 18 −
2, 45
5
· 1, 711 = 18, 84
X
1
X
2
X
1
∼ N(µ
1
, σ
2
1
); X
1
∼ N(µ
2
, σ
2
2
)
µ
1
− µ
2

σ
1
, σ
2

(
x
1
− x
2
) −

S
2
1
n
1
+
S
2
2
n
2
t
(k)
α/2
; (
x
1
− x

2
) +

S
2
1
n
1
+
S
2
2
n
2
t
(k)
α/2

x
1
=
330 + 360 + 400 + 350
4
= 360
x
2
=
290 + 320 + 340 + 370
4
= 330

s
2
1
=
1
3
(330
2
+ 360
2
+ 400
2
+ 350
2
− 4.360
2
) =
2600
3
⇒ s
1
= 29, 4392
s
2
2
=
1
3
(290
2

+ 320
2
+ 340
2
+ 370
2
− 4.330
2
) =
3400
3
⇒ s = 33, 665
1 − α = 0, 95 ⇒ α/2 = 0, 025
c =
s
2
1
n
1
s
2
1
n
1
+
s
2
2
n
2

=
2600
3.4
2600
3.4
+
3400
3.4
=
2600
6000
=
13
30
k =
(n
1
− 1)(n
2
− 1)
(n
2
− 1)c
2
+ (n
1
− 1)(1 − c)
2
=
3.3

3 ·

13
30

2
+ 3 ·

17
30

2
= 6
t
(k)
α/2
= t
(6)
0,025
= 2, 447

(360 − 330) −

2600
3.4
+
3400
3.4
· 2, 447; (360 − 330) +


2600
3.4
+
3400
3.4
· 2, 447

= (−24, 72; 84, 72)
X
1
X
2
X
1
∼ N(µ
1
, σ
2
1
); X
2
∼ N(µ
2
, σ
2
2
)
L = 90(µ
1
− µ

2
) − 80
X
1
, X
2
X
1
∼ N(µ
1
, σ
2
1
); X
2
∼ N(µ
2
, σ
2
2
) µ
1
, µ
2
, σ
2
1
, σ
2
2

µ
1
− µ
2
(1 − α) µ
1
− µ
2

(
X
1
− X
2
) −

S
2
1
n
1
+
S
2
2
n
2
t
(k)
α/2

; (
X
1
− X
2
) +

S
2
1
n
1
+
S
2
2
n
2
t
(k)
α/2

x
1
=
9 + 12 + 8 + 10 + 16
5
= 11
x
2

=
16 + 19 + 12 + 11 + 22
5
= 16
s
2
1
=
(9 − 11)
2
+ (12 − 11)
2
+ (8 − 11)
2
+ (10 − 11)
2
+ (16 − 11)
2
4
= 10
⇒ s
1
=
√
10 = 3, 16
s
2
2
=
(16 − 16)

2
+ (19 − 16)
2
+ (12 − 16)
2
+ (11 − 16)
2
+ (22 − 16)
2
4
= 21, 5
⇒ s
2
=

21, 5 = 4, 64
c =
s
2
1
n
1
s
2
1
n
1
+
s
2

2
n
2
=
10
5
10
5
+
21,5
5
= 0, 3175
k =
(n
1
− 1)(n
2
− 1)
(n
2
− 1)c
2
+ (n
1
− 1)(1 − c)
2
=
4.4
4.0, 3175
2

+ 4.0, 6825
2
= 7, 1
1 − α = 0, 95 ⇒ α/2 = 0, 025 ⇒ t
(k)
α/2
= t
(8)
0,025
= 2, 306
µ
1
− µ
2

(
x
1
− x
2
) −

s
2
1
n
1
+
s
2

2
n
2
t
(k)
α/2
; (
x
1
− x
2
) +

s
2
1
n
1
+
s
2
2
n
2
t
(k)
α/2

=


(11 − 16) −

10
5
+
21, 5
5
2, 306; (11 − 16) +

10
5
+
21, 5
5
2, 306

=(−10, 788; 0, 788)
X
1
∼ N(µ
1
, σ
2
1
)
X
2
∼ N(µ
1
, σ

2
2
)
σ
2
1
, σ
2
2
(1 − α) µ
1
− µ
2

(
X
1
− X
2
) −

S
2
1
n
1
+
S
2
2

n
2
t
(k)
α/2
; (
X
1
− X
2
) +

S
2
1
n
1
+
S
2
2
n
2
t
(k)
α/2

x
1
= 78

s
2
1
=
1
n
1
− 1

(x
1i
−
x
1
)
2
=
1
7
· 1805 = 275, 875
x
2
= 99
s
2
2
=
1
n
2

− 1

(x
2i
−
x
2
)
2
=
1
45
· 11520 = 256

s
2
1
n
1
+
s
2
2
n
2
=

257, 875
8
+

256
46
≈ 6, 148
1 − α = 0, 95 ⇒ α/2 = 0, 025
c =
s
2
1
n
1
s
2
1
n
1
+
s
2
2
n
1
=
257,875
8
257,875
8
+
256
46
= 0, 8528

k
(n
1
− 1)(n
2
− 1)
(n
2
− 1)c
2
+ (n
1
− 1)(1 − c)
2
= 9, 58 ≈ 10
t
(k)
α/2
= t
(10)
0,025
= 2, 228
µ
1
− µ
2

(
x
1

− x
2
) −

s
2
1
n
1
+
s
2
2
n
2
t
(k)
α/2
; (
x
1
− x
2
) +

s
2
1
n
1

+
s
2
2
n
2
t
(k)
α/2

=

(78 − 99) − 6, 148.2, 228; (78 − 99) + 6, 148.2, 228

= (−34, 7; −7, 3)
7, 3 < µ
2
− µ
1
< 34, 7
µ
1
, µ
2
µ
D
= µ
1
− µ
2

µ
D

D −
S
D
√
n
t
(n−1)
α/2
;
D +
S
D
√
n
t
(n−1)
α/2

⇒ 1 − α = 0, 95 ⇒ α/2 = 0, 025 ⇒ t
(n−1)
α/2
= t
(5)
0,025
= 2, 571
d
1

(d
i
− d) (d
i
− d)
2
d =
1
n

x
i
=
120
6
= 20
s
2
D
=
1
n − 1

(d
i
−
d)
2
=
1600

5
= 320 ⇒ s
D
=
√
320 = 17, 89
µ
D

d −
s
D
√
n
t
(n−1)
α/2
;
d +
s
D
√
n
t
(n−1)
α/2

=

20 −

17, 89
√
6
· 2, 571; 20 +
17, 89
√
6
· 2, 571

=(1, 22; 38, 78)
⇒ µ
D
= µ
2
− µ
1
µ
D
σ
2
µ
D

D −
S
√
n
t
(n−1)
α/2

;
D +
S
√
n
t
(n−1)
α/2

x
i
y
i
d
i
d
2
i
d =
1
n

d
i
=
9
5
= 1, 8
s
2

=
1
n − 1


d
2
i
− n.
d
i

=
1
4
(81 − 5.1, 8
2
) = 16, 2 ⇒ s = 4, 025
1 − α = 0, 95 ⇒ t
(n−1)
α/2
= t
(4)
0,025
= 2, 776
µ
D
= µ
2
− µ

1

d −
s
√
n
t
(n−1)
α/2
;
d +
s
√
n
t
(n−1)
α/2

=

1, 8 −
4, 025
√
5
· 2, 776; 1, 8 +
4, 025
√
5
· 2, 776


= (−3, 197; 6, 797)
X
1
, X
2
µ
1
− µ
2
σ
1
, σ
2
µ
1
− µ
2

(
X
1
− X
2
) −

S
2
1
n
1

+
S
2
2
n
2
t
(k)
α/2
; (
X
1
− X
2
) +

S
2
1
n
1
+
S
2
2
n
2
t
(k)
α/2


n
1
= n
2
= 30;
x
1
= 65, 3; x
2
= 60, 4
s
1
= 7; s
2
= 6

s
2
1
n
1
+
s
2
2
n
2
=


49
30
+
36
30
= 1, 6833
c =
s
2
1
n
1
s
2
1
n
1
+
s
2
2
n
2
=
49
30
49
30
+
36

30
= 0, 5765
k =
(n
1
− 1)(n
2
− 1)
(n
2
− 1)c
2
+ (n
1
− 1)
2
(1 − c)
2
=
29.29
29.0, 5765
2
+ 29.0, 4235
2
= 56, 67
k = 57; 1 − α = 0, 95 ⇒ t
(k)
α/2
= t
(57)

0,025
≈ u
0,025
= 1, 96

(
x
1
− x
2
) −

s
2
1
n
1
+
s
2
2
n
2
t
(k)
α/2
; (
x
1
− x

2
) +

s
2
1
n
1
+
s
2
2
n
2
t
(k)
α/2

=

(65, 3 − 60, 4) −1, 6833.1, 96; (65, 3 − 60, 4) −1, 6833.1, 96

= (1, 6; 8, 2)
X ∼ N(µ
1
, σ
1
= 0, 517); x = 1, 317; n
1
= 230

Y ∼ N(µ
2
, σ
2
= 0, 485);
y = 1, 512; n
2
= 302

σ
2
1
n
1
+
σ
2
2
n
2
=

0, 517
2
230
+
0, 485
2
302
= 0, 044057

1 − α = 0, 95 ⇒ u
α/2
= u
0,025
= 1, 96
µ
1
− µ
2
σ
2
1
, σ
2
2

(
x − y) −

σ
2
1
n
1
+
σ
2
2
n
2

u
α/2
; (
x − y) +

σ
2
1
n
1
+
σ
2
2
n
2
u
α/2

=

(1, 317 − 1, 512) − 0, 044057.1, 96; (1, 317 − 1, 512) + ·0, 044057.1, 96

= (−0, 28135; −0, 10865)
0, 10865 < µ
2
− µ
1
< 0, 28135
X ∼ N(µ

1
, σ
2
1
); n
1
= 300; x = 68, 1; s
1
= 8, 2
Y ∼ N(µ
2
, σ
2
2
); n
2
= 400;
y = 77, 3; s
2
= 5, 3

s
2
1
n
1
+
s
2
2

n
2
=

8, 2
2
300
+
5, 3
2
400
= 0, 543
c =
s
2
1
n
1
s
2
1
n
1
+
s
2
2
n
2
=

8,2
2
300
8,2
2
300
+
5,3
2
400
= 0, 76143
k =
(n
1
− 1)(n
2
− 1)
(n
2
− 1)
2
c
2
+ (n
1
− 1)
2
(1 − c)
2
=

299.399
399.0, 76143
2
+ 299.0, 23857
2
= 480, 4
k = 481; 1 − α = 0, 95 ⇒ t
(k)
α/2
= t
(481)
0,025
≈ u
0,025
= 1, 96

(
x − y) −

s
2
1
n
1
+
s
2
2
n
2

t
(k)
α/2
; (
x − y) +

s
2
1
n
1
+
s
2
2
n
2
t
(k)
α/2

=

(68, 1 − 77, 3) −0, 543.1, 96; (68, 1 −77, 3) −0, 543.1, 96

= (−10, 263; −8, 137)
8, 137 < µ
2
− µ
1

< 10, 263
X ∼ N(µ
1
, σ
2
1
); n
1
= 7; x = 17, 5; s
1
= 3, 2
Y ∼ N(µ
2
, σ
2
2
); n
2
= 11;
y = 15, 3; s
2
= 2, 9

s
2
1
n
1
+
s

2
2
n
2
=

3, 2
2
7
+
2, 9
2
11
= 1, 493
c =
s
1
1
n
1
s
2
1
n
1
+
s
2
2
n

2
=
3,2
2
7
3,2
2
7
+
2,9
2
11
= 0, 657
k =
(n
1
− 1)(n
2
− 1)
(n
2
− 1)c
2
+ (n
1
− 1)(1 − c)
2
=
6.10
10.0, 657

2
+ 6.0, 343
2
= 11, 95
k = 12; 1 − α = 0, 95 ⇒ t
(k)
α/2
= t
(12)
0,025
= 2, 179
µ
1
− µ
2

(
x − y) −

s
2
1
n
1
+
s
2
2
n
2

t
(k)
α/2
; (
x − y) +

s
2
1
n
1
+
s
2
2
n
2
t
(k)
α/2

=

(17, 5 − 15, 3) −1, 493.2, 179; (17, 5 − 15, 3) + 1, 493.2, 179

= (−1, 053; 5, 453)
X ∼ N(µ
1
, σ
2

1
); n
1
= 6; x =
14, 3
3
≈ 4, 7667; s
2
1
= 0, 4547
Y ∼ N(µ
2
, σ
2
2
); n
2
= 8;
y =
41, 7
8
= 5, 2125; s
2
2
= 0, 2984

s
2
1
n

1
+
s
2
2
n
2
= 0, 3363
c =
s
2
1
n
1
s
2
1
n
1
+
s
2
2
n
2
=
0,4547
6
0,4547
6

+
0,2984
8
= 0, 67
k =
(n
1
− 1)(n
2
− 1)
(n
2
− 1)c
2
+ (n
1
− 1)(1 − c)
2
=
5.7
7.0.67
2
+ 5.0.33
2
= 79, 5
k = 10; 1 − α = 0, 95 ⇒ t
(k)
α/2
= t
(10)

0,025
= 2, 228
µ
1
− µ
2

(
x − y) −

s
2
1
n
1
+
s
2
2
n
2
t
(k)
α/2
; (
x − y) +

s
2
1

n
1
+
s
2
2
n
2
t
(k)
α/2

=

(4, 7667 −5, 2125) − 0, 3363.2, 228; (4, 7667 − 5, 2125) −0, 3363.2, 228

= (−1, 195; 0, 303)

f −

f(1 − f)
√
n
u
α/2
; f +

f(1 −f)
√
n

u
α/2

f =
100−10
100
= 0, 9
1 − α = 0, 95 ⇒ α/2 = 0, 025 ⇒ u
α/2
= u
0,025
= 1, 96
′

0, 9 −
√
0, 9.0, 1
10
· 1, 96; 0, 9 +
√
0, 9.0, 1
10
· 1, 96

= (0, 84; 0, 96)
1 − α

f −

f(1 − f)

√
n
u
α/2
; f +

f(1 −f)
√
n
u
α/2

1 − α = 0, 09 ⇒ u
α
= u
0,025
= 1, 96
n = 1000; f =
860
1000
= 0, 86

f −

f(1 − f)
√
n
u
α/2
; f +


f(1 − f)
√
n
u
α/2

=

0, 86 −
√
0, 86.0, 14
√
1000
· 1, 96; 0, 86 +
√
0, 86.0, 14
√
1000
· 1, 96

= (0, 8385; 0, 8815)
1 − α

f −

f(1 − f)
√
n
u

α/2
; f +

f(1 −f)
√
n
u
α/2

1 − α = 0, 09 ⇒ u
α
= u
0,025
= 1, 96
n = 200; f =
8
200
= 0, 04

0, 04 −
√
0, 04.0, 96
√
200
· 1, 96; 0, 04 +
√
0, 04.0, 96
√
200
· 1, 96


= (0, 01284; 0, 067)
1 − α

f −

f(1 − f)
√
n
u
α/2
; f +

f(1 −f)
√
n
u
α/2

1 − α = 0, 09 ⇒ u
α
= u
0,025
= 1, 96
n = 200; f =
150
200
= 0, 75

0, 75 −

√
0, 75.0, 25
√
200
· 1, 96; 0, 75 +
√
0, 75.0, 25
√
200
· 1, 96

= (0, 69; 0, 81)
1 − α
f +
√
f(1−f)
√
n
u
α
n = 400; f =
20
400
= 0, 05
1 − α = 0, 95; u
α
= u
0,05
= 1, 64
p < f +


f(1 −f)
√
n
u
α
= 0, 05 +
√
0, 05.0, 95
√
400
· 1, 64 = 0, 06787
1 − α
f −
√
f(1−f)
√
n
u
α
n = 1600; f =
960
1600
= 0, 6
1 − α = 0, 99 ⇒ u
α
= u
0,01
= 2, 33
p > f −


f(1 −f)
√
n
u
α
= 0, 6 −
√
0, 6.0, 4
√
1600
· 2, 33 = 0, 5715
1 − α
f +
√
f(1−f)
√
n
u
α
n = 400; f =
160
400
= 0, 4
1 − α = 0, 95; u
α
= u
0,05
= 1, 64
p < f +


f(1 − f)
√
n
u
α
= 0, 05 +
√
0, 4.0, 6
√
400
· 1, 64 = 0, 44017
I
0
= 0, 02
⇒ f = 0, 9; u
α/2
= u
0,025
= 1, 96
n 

4f(1 − f)
I
2
0
u
2
α/2


=

4.0, 9.0, 1
0, 02
2
· 1, 96

= [3457, 44]
p
1
, p
2
p
1
− p
2
n
1
= 1500, n
2
= 1000 p
1
− p
2
1 − α = 0, 95

(f
1
− f
2

) − s
f
u
α/2
; (f
1
− f
2
) + s
f
u
α/2

f
1
= 0, 43; f
2
= 0, 38
s
f
=

0, 43.0, 57
1500
+
0, 38.0, 62
1000
= 0, 02
1 − α = 0, 95 ⇒ u
α/2

= u
0,025
= 1, 96
p
1
− p
2
((0, 43 − 0, 38) −0, 02.1, 96; (0, 43 − 0, 38) + 0, 02.1, 96) = (0, 01085; 0, 08915)
p
1
, p
2
p
1
− p
2
p
1
− p
2
1 − α

(f
1
− f
2
) − S
f
u
α/2

; (f
1
− f
2
) + S
f
u
α/2

n
1
= 100; f
1
=
7
100
= 0, 07
n
2
= 150; f
2
=
12
150
= 0, 08
1 − α = 0, 95 ⇒ u
α/2
= u
0,025
= 1, 96

s
f
=

f
1
(1 − f
1
)
n
1
+
f
2
(1 − f
2
)
n
2
=

0, 07.0, 93
100
+
0, 08.0, 92
150
= 0, 03379

(f
1

− f
2
) − s
f
u
α/2
; (f
1
− f
2
) + s
f
u
α/2

=

(0, 07 − 0, 08) −0, 03379.1, 96; (0, 07 −0, 08) + 0, 03379.1, 96

= (−0, 07623; 0, 05623)
p
1
, p
2
p
1
− p
2
p
1

− p
2
1 − α

(f
1
− f
2
) − S
f
u
α/2
; (f
1
− f
2
) + S
f
u
α/2

n
1
= 1000; f
1
=
30
1000
= 0, 03
n

2
= 800; f
2
=
15
800
= 0, 01875
1 − α = 0, 95 ⇒ u
α/2
= u
0,025
= 1, 96
s
f
=

0, 03.0, 97
1000
+
0, 01875.0, 98125
800
= 0, 0722

(f
1
− f
2
) − s
f
u

α/2
; (f
1
− f
2
) + s
f
u
α/2

=

(0, 03 − 0, 01875) −0, 0722.1, 96; (0, 03 − 0, 01875) −0, 0722.1, 96

= (−0, 0029; 0, 0254)
p
1
, p
2
p
1
−p
2
p
1
− p
2
1 − α

(f

1
− f
2
) − S
f
u
α/2
; (f
1
− f
2
) + S
f
u
α/2

n
1
= 162; f
1
=
61
162
= 0, 3765
n
2
= 189; f
2
=
106

189
= 0, 5608
1 − α = 0, 95 ⇒ u
α/2
= u
0,025
= 1, 96
s
f
=

0, 3765.0, 6235
162
+
0, 5608.0, 4392
189
= 0, 05246

(f
1
− f
2
) − s
f
u
α/2
; (f
1
− f
2

) + s
f
u
α/2

=

(0, 3765 −0, 5608) −0, 05246.1, 96; (0, 3765 − 0, 5608) + 0, 05246.1, 96)
= (−0, 2873; 0, 0815)
p
1
−p
2
p
1
− p
2
1 − α

(f
1
− f
2
) − S
f
u
α/2
; (f
1
− f

2
) + S
f
u
α/2

n
1
= 150; f
1
=
30
150
= 0, 2
n
2
= 140; f
2
=
17
140
= 0, 12143
1 − α = 0, 95 ⇒ u
α/2
= u
0,025
= 1, 96
s
f
=


0, 2.0, 8
150
+
0, 12143.0, 87857
140
= 0, 043

(f
1
− f
2
) − s
f
u
α/2
; (f
1
− f
2
) + s
f
u
α/2

=

(0, 2 − 0, 12143) − 0, 043.1, 96; (0, 2 − 0, 12143) + 0, 043.1, 96

= (−0, 00524; 0, 1624)

X ∼ N(µ, σ
2
)
µ, σ
2
σ
2
µ σ
2
1 − α

(n − 1)S
2
χ
2(n−1)
α/2
;
(n − 1)S
2
χ
2(n−1)
1−α/2
