14 trang 2 lượt tải

Bài giảng Chương 5. Ước lượng tham số môn Xác suất thống kê | Đại học Duy Tân

4

Bài giảng Chương 5. Ước lượng tham số môn Xác suất thống kê | Đại học Duy Tân. Tài liệu được sưu tầm giúp bạn tham khảo, ôn tập và đạt kết quả cao. Mời bạn đọc đón xem.

Môn: Xác xuất thống kê (STA 151) 144 tài liệu

Trường: Đại học Duy Tân 1.9 K tài liệu

Tác giả:

Profile

Xuân Lộc Nguyễn

4 ngày trước
Tải xuống
Báo cáo
Danh sách Quiz
Tải xuống
/14
❳⑩❈ ❙❯❻❚ ❚❍➮◆● ❑➊
❚!♥ ❚❤➜% ❚&
✣➔ ◆➤♥❣✱ ✷✵✶✾
❚!♥ ❚❤➜% ❚& ✶✴✹✵
❈❤"#♥❣ ✺✳ ()❝ ❧",♥❣ -❤❛♠ 01
✶✳ ,-❝ ❧01♥❣ ✤✐➸♠
✶✳✶ ✣6♥❤ ♥❣❤➽❛
✲ ❈❤♦ ♠➝✉ ♥❣➝✉ ♥❤✐➯♥
{X
1
, X
2
, ..., X
n
}
+, +-♥❣ +❤➸ ❝0 ♣❤➙♥ ♣❤3✐ ♣❤4 +❤✉5❝ ✈➔♦
+❤❛♠ 93
θ
❝❤:❛ ❜✐➳+✳ ❑❤✐ ✤0✱ ♠5+ ❤➔♠ ✭+❤3♥❣ ❦➯✮
ˆ
θ =
ˆ
θ(X
1
, ..., X
n
)
✤:D❝ ❣E✐ ❧➔ ♠5+
!❝ ❧ $♥❣
❝G❛ +❤❛♠ 93
θ
✳
✲ ❱I✐ ♠5+ ♠➝✉ ❣✐→ +KL ❝4 +❤➸
{x
1
, x
2
, ..., x
n
}
+❛ +❤✉ ✤:D❝ ♠5+ ❣✐→ +KL ❝4 +❤➸ ❝G❛
ˆ
θ
✳
❑❤✐ ✤0✱ ❣✐→ +KL
ˆ
θ(x
1
, ..., x
n
)
✤:D❝ ❣E✐ ❧➔
!❝ ❧ $♥❣ ✤✐➸♠
❝G❛ +❤❛♠ 93
θ
❞N❛ +K➯♥ ♠➝✉ ❣✐→
+KL
{x
1
, x
2
, ..., x
n
}
✳
❚!♥ ❚❤➜% ❚& ✷✴✹✵
✶✳✷ :❤➙♥ ❧♦↕✐ 0-❝ ❧01♥❣
✲ OI❝ ❧:D♥❣
ˆ
θ =
ˆ
θ(X
1
, ..., X
n
)
✤:D❝ ❣E✐ ❧➔
!❝ ❧ $♥❣ ❦❤-♥❣ ❝❤➺❝❤
❝G❛ +❤❛♠ 93
θ
♥➳✉
E(
ˆ
θ) = θ
✳
❚K♦♥❣ +K:Q♥❣ ❤D♣ ♥❣:D❝ ❧↕✐ +❤➻ +❛ ❣E✐
ˆ
θ
✤:D❝ ❣E✐ ❧➔
!❝ ❧ $♥❣ ❝❤➺❝❤
✈➔ ❣✐→ +KL
b(θ) =
E(
ˆ
θ) −θ
✤:D❝ ❣E✐ ❧➔
✤/ ❝❤➺❝❤ ❝0❛ !❝ ❧ $♥❣
✳
✲ OI❝ ❧:D♥❣
ˆ
θ =
ˆ
θ(X
1
, ..., X
n
)
✤:D❝ ❣E✐ ❧➔
!❝ ❧ $♥❣ ❦❤-♥❣ ❝❤➺❝❤ 2✐➺♠ ❝➟♥
❝G❛ +❤❛♠
93
θ
♥➳✉
lim
n→+∞
E(
ˆ
θ) = θ.
✲ ❈❤♦
ˆ
θ
1
✈➔
ˆ
θ
2
❧➔ ❤❛✐ :I❝ ❧:D♥❣ ❦❤T♥❣ ❝❤➺❝❤ ❝G❛ +❤❛♠ 93
θ
✳ ❚❛ ♥0✐ :I❝ ❧:D♥❣
ˆ
θ
1
❤✐➺✉
5✉↔ ❤7♥
:I❝ ❧:D♥❣
ˆ
θ
2
♥➳✉
D(
ˆ
θ
1
) < D(
ˆ
θ
2
)
✳
✲ OI❝ ❧:D♥❣
ˆ
θ
❝G❛
θ
❧➔ :I❝ ❧:D♥❣ ❦❤T♥❣ ❝❤➺❝❤ ✈➔ ❝0 ♣❤:V♥❣ 9❛✐
D(
ˆ
θ)
❜➨ ♥❤➜+ ✤:D❝
❣E✐ ❧➔
!❝ ❧ $♥❣ 282 ♥❤➜2
✳
❚!♥ ❚❤➜% ❚& ✸✴✹✵
!❝ ❧$%♥❣ ❦❤*♥❣ ❝❤➺❝❤ ❝,❛ ./✉♥❣ ❜➻♥❤ ✈➔ ♣❤$6♥❣ 7❛✐
✣:♥❤ ❧;✿
❈❤♦ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥
X
❝,❛ ♠/0 01♥❣ 0❤➸ ❝3
E(X) = µ, D(X) = σ
2
✳ ●✐↔ 78
{X
1
, X
2
, ..., X
n
}
❧➔ ♠➝✉ ♥❣➝✉ ♥❤✐➯♥ ❝,❛
X
✳ ❑❤✐ ✤3✿
✐✮
X =
1
n
n
P
i=1
X
i
❧➔ ?@❝ ❧?A♥❣ ❦❤C♥❣ ❝❤➺❝❤ ❝,❛
µ
✳
✐✐✮
S
2
=
1
n − 1
n
P
i=1
(X
i
−
X)
2
❧➔ ?@❝ ❧?A♥❣ ❦❤C♥❣ ❝❤➺❝❤ ❝,❛
σ
2
✳
◆❤➟♥ ①➨.✿
❈3 0❤➸ ❝❤E♥❣ ♠✐♥❤ ✤?A❝ F➡♥❣✿
E(S
2
∗
) =
n − 1
n
σ
2
✈@✐
S
2
∗
=
1
n
n
X
i=1
(X
i
−
¯
X)
2
✣✐➲✉ ♥➔② ❝3 ♥❣❤➽❛
S
2
∗
❧➔ ?@❝ ❧?A♥❣ ❝❤➺❝❤ ✭❦❤C♥❣ ❝❤➺❝❤ 0✐➺♠ ❝➟♥✮ ❝,❛
σ
2
✱ ❞♦ ✤3 ♥3
✓❦❤C♥❣ ✤?A❝ ❝❤R♥✔ ❧➔♠ ♣❤?U♥❣ 7❛✐ ♠➝✉✳
❚!♥ ❚❤➜% ❚& ✹✴✹✵
!❝ ❧$%♥❣ ❦❤*♥❣ ❝❤➺❝❤ ❝,❛ .➾ ❧➺
✣:♥❤ ❧;✿
❈❤♦ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥
X
❝3 ♣❤➙♥ ♣❤W✐ ❇❡F♥♦✉❧❧✐ ✈@✐ 0❤❛♠ 7W
p
✳ ●R✐
(X
1
, X
2
, ..., X
n
)
❧➔ ♠➝✉ ♥❣➝✉ ♥❤✐➯♥ ❝,❛
X
✳ ❑❤✐ ✤3✿
ˆ
P =
X
1
+ X
2
+ ... + X
n
n
❧➔ ♠/0 ?@❝ ❧?A♥❣ ❦❤C♥❣ ❝❤➺❝❤ ❝,❛ 0❤❛♠ 7W
p
✳
B ♥❣❤➽❛✿
◆➳✉ ❞➜✉ ❤✐➺✉
A
❝3 0➾ ❧➺
p
❝❤?❛ ❜✐➳0✱ ✈➔ 0F♦♥❣ ♠➝✉ ✤✐➲✉ 0F❛ ❦➼❝❤ 0❤?@❝
n
❝3
k
❝→ 0❤➸ ♠❛♥❣ ❞➜✉ ❤✐➺✉
A
✱ 0❤➻ 0➛♥ 7✉➜0
f
n
= k/n
❧➔ ♠/0 ?@❝ ❧?A♥❣ ❦❤C♥❣ ❝❤➺❝❤ ❝,❛
p
✳
❚!♥ ❚❤➜% ❚& ✺✴✹✵
❱➼ ❞G ✶
✲ ❈❤♦ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥
X
❝- ❦/ ✈1♥❣
µ
✈➔ ♣❤45♥❣ 6❛✐
σ
2
❝❤4❛ ❜✐➳8✳ ✣➸ 4<❝ ❧4>♥❣
❝❤♦
µ
✈➔
σ
2
♥❣4?✐ 8❛ 8✐➳♥ ❤➔♥❤ ✤✐➲✉ 8B❛ ♠➝✉ ❦➼❝❤ 8❤4<❝ ✷✵✵ ✈➔ 8➼♥❤ ✤4>❝ 8B✉♥❣ ❜➻♥❤ ✈➔
♣❤45♥❣ 6❛✐ ❝❤✉➞♥ ♠➝✉✿
¯x = 125, 8; s
2
= 2, 76
❑❤✐ ✤-✱ 8❛ ❝- 8❤➸ ①❡♠
µ ≈ 125, 8
✈➔
σ
2
≈ 2, 76
✳
✲ ✣➸ 4<❝ ❧4>♥❣ 8➾ ❧➺
p
❝P 8B✐ Q♥❣ ❤R ❝❤♦ S♥❣ ❝P ✈✐➯♥ ❆✱ ♥❣4?✐ 8❛ ❦❤↔♦ 6→8 ♥❣➝✉ ♥❤✐➯♥
4000
♥❣4?✐ 8❤➻ ❝- ✷✻✹✵ ♥❣4?✐ Q♥❣ ❤R S♥❣ ❝P ✈✐➯♥ ♥➔②✳ ◆❤4 ✈➟②✱ 8❛ ❝- 8❤➸ ①❡♠ 8➾ ❧➺ Q♥❣
❤R S♥❣ ❝P ✈✐➯♥ ❆ ①➜♣ ①➾✿
p ≈ 2640/4000 = 0, 66
❚!♥ ❚❤➜% ❚& ✻✴✹✵
X θ
{X
1
, X
2
, ..., X
n
}
L = L(X
1
, ..., X
n
) U = U (X
1
, ..., X
n
)
α ∈ (0, 1)
P (L < θ < U) = 1 − α.
(L, U) θ 1 − α
α U − L
(L, U) θ
1 − α
1 − α
α
1
, α
2
∈ (0, 1) α
1
+ α
2
= α
G
n
= G
n
(X
1
, ..., X
n
, θ)
G
n
G
n
n → ∞ θ G
n
θ
a b P (G
n
≤ a) = α
1
P (G
n
< b) = 1 − α
2
a < G
n
(X
1
, ..., X
n
, θ) < b θ
L, U
P (L < θ < U) ≥ 1 − α.
α
1
= 0 α
2
= 0
✷✳✷ ❑❤♦↔♥❣ (✐♥ ❝➟② ❝❤♦ ❦➻ ✈0♥❣ ❦❤✐ ✤➣ ❜✐➳( ♣❤67♥❣ 8❛✐
❇➔✐ (♦→♥
✳ ❈❤♦ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥
X
❝-❛ ♠01 12♥❣ 1❤➸ ❝4 ♣❤➙♥ ♣❤7✐ ❝❤✉➞♥
N(µ; σ
2
)
✈:✐
µ
❝❤;❛ ❜✐➳1 ✈➔
σ
2
✤➣ ❜✐➳1✳ ❚➻♠ ❦❤♦↔♥❣ 1✐♥ ❝➟② ❝❤♦
µ
✈:✐ ♠E❝ F ♥❣❤➽❛
α
✳
●>✐ ?
✳ ◆➳✉
{X
1
, X
2
, ..., X
n
}
❧➔ ♠➝✉ ♥❣➝✉ ♥❤✐➯♥ ❝-❛
X
1❤➻
Z =
X − µ
σ/
√
n
∼ N(0; 1).
✲ ❱:✐
α ∈ (0; 1)
✱ ❝❤M♥
α
1
= α
2
= α /2
✈➔
z
α/2
= Φ
−1
(1 −
α
2
)
❧➔
❣✐→ (@A (B✐ ❤↕♥
♠E❝
α/2
❝-❛ ♣❤➙♥ ♣❤7✐ ❝❤✉➞♥ 1➢❝✱ ❣✐↔✐ ❜➜1 ♣❤;P♥❣ 1Q➻♥❤✿
−z
α/2
< Z < z
α/2
⇔
X − z
α/2
σ
√
n
< µ <
X + z
α/2
σ
√
n
1❛ ✤;S❝ ❦❤♦↔♥❣ 1✐♥ ❝➟② ✤7✐ ①E♥❣✳
✲ ◆➳✉ ❝❤M♥
α
1
= 0
❤♦➦❝
α
2
= 0
✳ ▲W❝ ♥➔② 1❛ 1❤❛②
z
α/2
❜X✐
z
α
✈➔ 1❤✉ ✤;S❝ ❦❤♦↔♥❣ 1✐♥
❝➟② ♠01 ♣❤➼❛✳
❚!♥ ❚❤➜% ❚& ✶✵✴✹✵
❚!♥ ❚❤➜% ❚& ✶✶✴✹✵
❛✳ ❑➳( D✉↔
❱:✐ ✤0 1✐♥ ❝➟②
1 − α
✿
✲ ❑❤♦↔♥❣ 1✐♥ ❝➟② ✤7✐ ①E♥❣ ❝-❛
µ
✿
x − z
α/2
σ
√
n
< µ <
x + z
α/2
σ
√
n
,
✲ ❑❤♦↔♥❣ 1✐♥ ❝➟② 17✐ ✤❛ ❝-❛
µ
✿
µ <
x + z
α
σ
√
n
.
✲ ❑❤♦↔♥❣ 1✐♥ ❝➟② 17✐ 1❤✐➸✉ ❝-❛
µ
✿
µ >
x − z
α
σ
√
n
.
❚!♥ ❚❤➜% ❚& ✶✷✴✹✵
❱➼ ❞# ✷
❑❤"✐ ❧%&♥❣ ✭❦❣✮ ❝-❛ ♠01 1❤✐➳1 ❜4 ❝5 ♣❤➙♥ ♣❤"✐ ❝❤✉➞♥
N(µ; σ
2
)
✈;✐
σ = 0. 2
✭❦❣✮✳ ❈❤>♥
♥❣➝✉ ♥❤✐➯♥ ✷✺ 1❤✐➳1 ❜4 ♥❣%C✐ 1❛ 1➼♥❤ ✤%&❝ 1F✉♥❣ ❜➻♥❤ ♠➝✉
x = 65, 1
✭❦❣✮✳ ❱;✐ ✤0 1✐♥ ❝➟②
✾✺✪ ❤➣② 1➻♠ ❦❤♦↔♥❣ 1✐♥ ❝➟② ✭✤"✐ ①Q♥❣✮ ❝❤♦ ❦❤"✐ ❧%&♥❣ 1F✉♥❣ ❜➻♥❤ ❝-❛ 1❤✐➳1 ❜4 ♥➔②✳ ❈❤♦
❜✐➳1
z
0,025
= 1, 96
✳
●✐↔✐✳
✲ ❈→❝ $% ✤➦❝ ()*♥❣ ♠➝✉✿
n = 25; ¯x = 65, 1
✳
✲ ✣3 (✐♥ ❝➟②✿
1 − α = 0, 95 ⇒ α = 0, 05; z
α/2
= z
0,025
= 1, 96
✳
✲ ❙❛✐ $% *9❝ ❧*;♥❣✿
ε = z
α/2
σ
√
n
= 1, 96 ∗
0, 2
√
25
= 0, 0784
✲ ❑❤♦↔♥❣ (✐♥ ❝➟② ❝❤♦ ❦❤%✐ ❧*;♥❣ ()✉♥❣ ❜➻♥❤
µ
❝C❛ (❤✐➳( ❜E ♥➔②✿
¯x − ε < µ < ¯x + ε ⇔ 65, 02 < µ < 65, 18
❚!♥ ❚❤➜% ❚& ✶✸✴✹✵
❜✳ ❱➜♥ ✤➲ ❝/ ♠➝✉
❚H ❝I♥❣ (❤J❝ ❦❤♦↔♥❣ (✐♥ ❝➟② ❝❤♦
µ
(❛ (❤➜② )➡♥❣ $❛✐ $% ❝C❛ *9❝ ❧*;♥❣
|x − µ|
❜➨ ❤N♥
❤♦➦❝ ❜➡♥❣
z
α/2
σ
√
n
✳
✲ ❙❛✐ $% *9❝ ❧*;♥❣✿
ε = z
α/2
σ
√
n
✲ ✣✐➲✉ ❦✐➺♥ ❤↕♥ ❝❤➳✿
ε < ∆
✈9✐
∆ > 0
❝❤♦ ()*9❝✳
ε = z
α/2
σ
√
n
< ∆
✲ ●✐↔✐ ❜➜( ♣❤*N♥❣ ()➻♥❤ (❤❡♦
n
✿
n >
z
α/2
σ
∆
2
.
❚!♥ ❚❤➜% ❚& ✶✹✴✹✵
❱➼ ❞# ✸
❚FT ❧↕✐ ✈;✐ ❱➼ ❞W ✷ ♥➳✉ ②➯✉ ❝➛✉ Y❛✐ Y" %;❝ ❧%&♥❣ ❦❤Z♥❣ ✈%&1 [✉→ ✵✱✵✺ 1❤➻ ✈;✐ ✤0 1✐♥ ❝➟②
✾✽✪ 1❛ ❝➛♥ ❝❤>♥ 1"✐ 1❤✐➸✉ ❜❛♦ ♥❤✐➯✉ 1❤✐➳1 ❜4 ✤➸ ❦❤↔♦ Y→1❄ ❈❤♦ ❜✐➳1
z
0,01
= 2, 326
✳
●✐↔✐✳
✲ ❙❛✐ $% *9❝ ❧*;♥❣✿
ε = z
α/2
σ
√
n
✲ ✣✐➲✉ ❦✐➺♥ ❤↕♥ ❝❤➳✿
ε < ∆ = 0, 05
✳
✲ ❙✉② )❛✿
n >
z
α/2
σ
∆
2
.
✲ ✣3 (✐♥ ❝➟②✿
1 − α = 0, 98 ⇒ α = 0, 2; z
α/2
= z
0,01
= 2, 326
✳
❉♦ ✤W✿
n >
2, 326 ∗ 0, 2
0, 05
2
= 86, 56
❱➟②✱ ❦➼❝❤ (❤*9❝ ♠➝✉ (%✐ (❤✐➸✉ ❝➛♥ ❦❤↔♦ $→(✿
n = 87
❚!♥ ❚❤➜% ❚& ✶✺✴✹✵
✷✳✸ ❑❤♦↔♥❣ )✐♥ ❝➟② ❝❤♦ ❦➻ ✈1♥❣ ❦❤✐ ❝❤2❛ ❜✐➳) ♣❤27♥❣ 8❛✐
❇➔✐ )♦→♥
✿ ❈❤♦ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥
X
❝-❛ ♠01 12♥❣ 1❤➸ ❝4 ♣❤➙♥ ♣❤7✐ ❝❤✉➞♥
N(µ; σ
2
)
✈:✐
µ
✈➔
σ
2
❝❤<❛ ❜✐➳1✳ ❚➻♠ ❦❤♦↔♥❣ 1✐♥ ❝➟② ❝❤♦
µ
✈:✐ ✤0 1✐♥ ❝➟②
1 − α
✳
●=✐ >
✳ ◆➳✉
{X
1
, X
2
, ..., X
n
}
❧➔ ♠➝✉ ♥❣➝✉ ♥❤✐➯♥ ❝-❛
X
1❤➻
T =
X − µ
S/
√
n
∼ T
n−1
✲ ❱:✐ ♠I❝ J ♥❣❤➽❛
α
✱ ❝❤M♥
α
1
= α
2
= α/2
✈➔ ❧➜② ❣✐→ 1PQ
t
n−1;α/2
❧➔
❣✐→ )?@ )A✐ ❤↕♥
❝-❛
♣❤➙♥ ♣❤7✐
T
n−1
✱ 1❛ ✤<R❝ ❦❤♦↔♥❣ 1✐♥ ❝➟② ✤7✐ ①I♥❣✳
✲ ❈❤M♥
α
1
= 0
❤♦➦❝
α
2
= 0
✳ ▲V❝ ♥➔② 1❛ 1❤❛②
t
n−1;α/2
❜W✐
t
n−1;α
✈➔ 1❤✉ ✤<R❝ ❦❤♦↔♥❣
1✐♥ ❝➟② ♠01 ♣❤➼❛✳
❚!♥ ❚❤➜% ❚& ✶✻✴✹✵
❚!♥ ❚❤➜% ❚& ✶✼✴✹✵
❑➳) C✉↔
❱:✐ ✤0 1✐♥ ❝➟②
1 − α
✿
✲ ❑❤♦↔♥❣ 1✐♥ ❝➟② ✤7✐ ①I♥❣ ❝-❛
µ
✿
x − t
n−1;α/2
s
√
n
< µ <
x + t
n−1;α/2
s
√
n
,
✲ ❑❤♦↔♥❣ 1✐♥ ❝➟② 17✐ ✤❛ ❝-❛
µ
✿
µ <
x + t
n−1;α
s
√
n
.
✲ ❑❤♦↔♥❣ 1✐♥ ❝➟② 17✐ 1❤✐➸✉ ❝-❛
µ
✿
x − t
n−1;α
s
√
n
< µ.
❚!♥ ❚❤➜% ❚& ✶✽✴✹✵
◆❤➟♥ ①➨&✿
✐✮ ❑❤✐
n > 30
✿
t
n−1;α/2
≈ z
α/2
✳
✐✐✮ ❑❤✐ ❦➼❝❤ )❤*+❝ ♠➝✉ ❧+♥ ✭
n > 30
✮ )❤❡♦ ✤5♥❤ ❧6 ❣✐+✐ ❤↕♥ )9✉♥❣ )➙♠ ❦➳) <✉↔ )9➯♥ ✈➝♥
→♣ ❞C♥❣ ✤*D❝ ❞E )❤✐➳✉ ❣✐↔ )❤✐➳) ✈➲ ✤✐➲✉ ❦✐➺♥ ♣❤➙♥ ♣❤H✐ ❝❤✉➞♥ ❝J❛ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥
X
✳
❱➼ ❞+ ✹
▼!" ♠➝✉ ✶✻ ♣✐♥ ❞,♥❣ ❝❤♦ 1♠❛3"♣❤♦♥❡ ✤67❝ ❝❤8♥ ♥❣➝✉ ♥❤✐➯♥ ❝:❛ ❝;♥❣ "② ❆ ❝> "✉?✐ "❤8
"3✉♥❣ ❜➻♥❤ "➼♥❤ "3➯♥ ♠➝✉
x = 24308
✭❣✐D✮ ✈➔ ✤! ❧➺❝❤ ❝❤✉➞♥ ♠➝✉
s = 727
✭❣✐D✮✳ ●✐↔ 1N
3➡♥❣ "✉?✐ "❤8 ♣✐♥ 1♠❛3"♣❤♦♥❡ ❝> ♣❤➙♥ ♣❤Q✐ ❝❤✉➞♥✳ ❱S✐ ✤! "✐♥ ❝➟② ✾✺✪✱ ❤➣② "➻♠ ❦❤♦↔♥❣
"✐♥ ❝➟② ❝❤♦ "✉?✐ "❤8 "3✉♥❣ ❜➻♥❤ ❝:❛ ♣✐♥ 1♠❛3"♣❤♦♥❡ ✤67❝ 1↔♥ ①✉➜" ❜]✐ ❝;♥❣ "② ❆✳ ❈❤♦
❜✐➳"
t
15;0,025
= 2, 1314
✳
❚!♥ ❚❤➜% ❚& ✶✾✴✹✵
●✐↔✐✳
❚❤❡♦ ❣✐↔ )❤✐➳)✿
n = 16; ¯x = 24308; s = 727
✣O )✐♥ ❝➟②✿
1 − α = 0, 95 ⇒ α = 0, 05; t
n−1;α/2
= t
15;0,025
= 2, 1314
❙❛✐ SH✿
ε = t
n−1;α/2
s
√
n
= 2, 1314 ∗
727
√
16
= 387, 382
❑❤♦↔♥❣ )✐♥ ❝➟② ❝❤♦ )✉T✐ )❤U )9✉♥❣ ❜➻♥❤ ❝J❛ ♣✐♥ S♠❛9)♣❤♦♥❡✿
¯x − ε < µ < ¯x + ε ⇔ 23920.618 < µ < 24695.382
❚!♥ ❚❤➜% ❚& ✷✵✴✹✵
❱➼ ❞+ ✺
◆➠♥❣ 1✉➜" ✭"↕✴❤❛✮ ❝:❛ ♠!" ❧♦↕✐ ❝➙② "3d♥❣ "✉➙♥ "❤❡♦ ❧✉➟" ♣❤➙♥ ♣❤Q✐ ❝❤✉➞♥✳ ❍➣② "➻♠ 6S❝
❧67♥❣ ❦❤♦↔♥❣ ✤Q✐ ①f♥❣ ♥➠♥❣ 1✉➜" "3✉♥❣ ❜➻♥❤ ❝:❛ ♠!" ❧♦↕✐ ❝➙② "3d♥❣ ♥➔② ✈S✐ ✤! "✐♥ ❝➟②
✾✺✪ "3➯♥ ❝g 1] ❜↔♥❣ 1Q ❧✐➺✉ ✤✐➲✉ "3❛ 1❛✉ ✤➙②✿
◆➠♥❣ 1✉➜"✭"↕✴❤❛✮ ✹✷✲✹✼ ✹✼✲✺✷ ✺✷✲✺✼ ✺✼✲✻✷ ✻✷✲✻✼
❙Q ✤✐➸♠ "❤✉ ❤♦↕❝❤ ✷ ✺ ✶✹ ✶✵ ✺
❈❤♦ ❜✐➳"
z
0,05
= 1, 645; z
0,025
= 1, 96; z
0,02
= 2, 054; z
0,01
= 2, 326
✳
●✐↔✐✳
❉↕♥❣ )❤✉ ❣U♥✿
◆➠♥❣ S✉➜)✭)↕✴❤❛✮ ✹✹✱✺ ✹✾✱✺ ✺✹✱✺ ✺✾✱✺ ✻✹✱✺
❙H ✤✐➸♠ )❤✉ ❤♦↕❝❤ ✷ ✺ ✶✹ ✶✵ ✺
❚!♥ ❚❤➜% ❚& ✷✶✴✹✵
❈→❝ #$ ✤➦❝ '()♥❣ ♠➝✉✿
n = 36; ¯x = 56, 028; s = 5, 321
✣1 '✐♥ ❝➟②✿
1 − α = 0, 95 ⇒ α = 0, 05; t
n−1;α/2
≈ z
α/2
= z
0,025
= 1, 96
❙❛✐ #$✿
ε = t
n−1;α/2
s
√
n
= 1, 96 ∗
5, 321
√
36
= 1, 738
❑❤♦↔♥❣ '✐♥ ❝➟② ❝❤♦ ♥➠♥❣ #✉➜' '(✉♥❣ ❜➻♥❤✿
¯x − ε < µ < ¯x + ε ⇔ 54, 29 < µ < 57, 766
❚!♥ ❚❤➜% ❚& ✷✷✴✹✵
✷✳✹ ❑❤♦↔♥❣ )✐♥ ❝➟② ❝❤♦ ♣❤/0♥❣ 1❛✐ ✭✤5❝ )❤➯♠✮
❇➔✐ )♦→♥
✿ ❈❤♦ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥
X
❝A❛ ♠1' 'B♥❣ '❤➸ ❝D ♣❤➙♥ ♣❤$✐ ❝❤✉➞♥
N(µ; σ
2
)
✈I✐
µ
✈➔
σ
2
❝❤)❛ ❜✐➳'✳ ❚➻♠ ❦❤♦↔♥❣ '✐♥ ❝➟② ❝❤♦
σ
2
✈I✐ ✤1 '✐♥ ❝➟②
1 − α
✳
❑➳) =✉↔✿
◆➳✉
{X
1
, X
2
, ..., X
n
}
❧➔ ♠➝✉ ♥❣➝✉ ♥❤✐➯♥ ❝A❛
X
'❤➻ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥✿
χ
2
=
(n − 1)S
2
σ
2
∼ χ
2
n−1
.
✲ ❱I✐ ♠R❝ S ♥❣❤➽❛
α
✱ ❝❤V♥
α
1
= α
2
= α/2
✈➔ ❧➜② ❣✐→ '(W
χ
2
n−1;α/2
, χ
2
n−1;1−α/2
❧➔
❣✐→ )@A
)B✐ ❤↕♥
♠R❝
α/2
✈➔
1 − α/2
❝A❛ ♣❤➙♥ ♣❤$✐
χ
2
n−1
✱ '❛ ✤)X❝ ❦❤♦↔♥❣ '✐♥ ❝➟②✳
✲ ❈❤V♥
α
1
= 0
❤♦➦❝
α
2
= 0
✳ ▲Z❝ ♥➔② '❛ '❤❛②
χ
2
n−1;α/2
❜[✐
χ
2
n−1;α
✱
χ
2
n−1;1−α/2
❜[✐
χ
2
n−1;1−α
✈➔ '❤✉ ✤)X❝ ❦❤♦↔♥❣ '✐♥ ❝➟② ♠1' ♣❤➼❛✳
❚!♥ ❚❤➜% ❚& ✷✸✴✹✵
❚!♥ ❚❤➜% ❚& ✷✹✴✹✵
❑➳" #✉↔
❱!✐ ✤$ %✐♥ ❝➟②
1 − α
✿
✲ ❑❤♦↔♥❣ %✐♥ ❝➟② ❤❛✐ ♣❤➼❛ ❝❤♦
σ
2
✿
(n − 1)s
2
χ
2
n−1;α/2
< σ
2
<
(n − 1)s
2
χ
2
n−1;1−α/2
,
✲ ❑❤♦↔♥❣ %✐♥ ❝➟② %4✐ ✤❛ ❝❤♦ ♣❤56♥❣ 7❛✐
σ
2
✿
0 < σ
2
<
(n − 1)s
2
χ
2
n−1;1−α
.
✲ ❑❤♦↔♥❣ %✐♥ ❝➟② %4✐ %❤✐➸✉ ❝❤♦ ♣❤56♥❣ 7❛✐
σ
2
✿
(n − 1)s
2
χ
2
n−1;α
< σ
2
< +∞.
❚!♥ ❚❤➜% ❚& ✷✺✴✹✵
❱➼ ❞) ✻
▼!" ❞➙② ❝❤✉②➲♥ "+ ✤!♥❣ ✤.♥❣ ❣↕♦ ✈➔♦ ❜❛♦ ✤56❝ ❦✐➸♠ ";❛✳ ❈➙♥ "❤> ✷✺ ❜❛♦ ❣↕♦ ❞♦ ❞➙②
❝❤✉②➲♥ ♥➔② "❤+❝ ❤✐➺♥ "❤➻ "➼♥❤ ✤56❝ ✤! ❧➺❝❤ ❝❤✉➞♥ ❦❤F✐ ❧56♥❣ ❣↕♦ ♠G✐ ❜❛♦
s = 0, 15
✭❦❣✮✳ ❱K✐ ✤! "✐♥ ❝➟② ✾✺✪ "➻♠ ❦❤♦↔♥❣ "✐♥ ❝➟② ❝❤♦ ♣❤5Q♥❣ R❛✐ ❝S❛ ❦❤F✐ ❧56♥❣ ❣↕♦ ✤56❝
✤.♥❣ ❝❤♦ ♠G✐ ❜❛♦ ❜T✐ ❞➙② ❝❤✉②➲♥ ♥➔②✳ ●✐↔ "❤✐➳" ❦❤F✐ ❧56♥❣ ❣↕♦ ✤56❝ ✤.♥❣ ❝❤♦ ♠G✐ ❜❛♦
"✉➙♥ "❤❡♦ ❧✉➟" ♣❤➙♥ ♣❤F✐ ❝❤✉➞♥✳ ❈❤♦ ❜✐➳"✿
χ
2
24;0.025
= 39, 364
✱
χ
2
24;0.975
= 12, 401.
✣→♣ ./✿
0, 0137 < σ
2
< 0, 0435
✳
❚!♥ ❚❤➜% ❚& ✷✻✴✹✵
✷✳✺ ❑❤♦↔♥❣ "✐♥ ❝➟② ❝❤♦ "✛ ❧➺
❇➔✐ "♦→♥✿
❳➨% ❞➜✉ ❤✐➺✉
A
%@♦♥❣ %A♥❣ %❤➸ ❝B %➾ ❧➺ ❧➔
p
✭❝❤5❛ ❜✐➳%✮✳ ✣➸ ❦❤↔♦ 7→% ❣✐→ %@M
p
%❛ %✐➳♥ ❤➔♥❤ ✤✐➲✉ %@❛ ♠$% ♠➝✉ ❦➼❝❤ %❤5!❝
n
✳ ●✐↔ 7R %@♦♥❣ ♠➝✉ ♥➔② ❝B
k
♣❤➛♥ %R ♠❛♥❣
❞➜✉ ❤✐➺✉
A
✳ ❍➣② %➻♠ ❦❤♦↔♥❣ %✐♥ ❝➟② ❝❤♦
p
✳
❑➳" #✉↔✿
✲ ●W✐
X
❧➔ 74 ♣❤➛♥ %R ♠❛♥❣ ❞➜✉ ❤✐➺✉
A
%@♦♥❣ ♠➝✉ ❦➼❝❤ %❤5!❝
n
✱ %❛ ❝B
X ∼ B(n, p)
✳
❚❤❡♦ ✤M♥❤ ❧[ ❣✐!✐ ❤↕♥ %➼❝❤ ♣❤➙♥ ▼♦✐✈@❡✲▲❛♣❧❛❝❡✱ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥✿
Z =
X − np
p
np(1 − p)
❝B ♣❤➙♥ ♣❤4✐ ①➜♣ ①➾ ❝❤✉➞♥ %➢❝
N(0, 1)
✳
✲ ❱!✐
α ∈ (0; 1)
❝❤♦ %@5!❝✱ ❝❤W♥
α
1
= α
2
= α/2
%❛ ✤5e❝ ❦❤♦↔♥❣✿
ˆp −z
α/2
r
p(1 − p)
n
< p < ˆp + z
α/2
r
p(1 − p)
n
✈!✐
ˆp = X/n = k/n
✳
❚!♥ ❚❤➜% ❚& ✷✼✴✹✵
❚!♥ ❚❤➜% ❚& ✷✽✴✹✵
▼➦" ❦❤→❝✱ ❞♦
ˆp
❧➔ ♠-" ./❝ ❧.0♥❣ ❝3❛
p
♥➯♥ "❛ ❝6 "❤➸ ①❡♠
p ≈ ˆp
❦❤✐
n
❧/♥✳
✣✐➲✉ ❦✐➺♥✿
nˆp > 5
✈➔
n(1 − ˆp) > 5
✳
❛✳ ❑➳, -✉↔
❱/✐ ✤- "✐♥ ❝➟②
1 − α
✿
✲ ❑❤♦↔♥❣ "✐♥ ❝➟② ✤E✐ ①F♥❣ ❝❤♦
p
✿
ˆp − z
α/2
r
ˆp(1 − ˆp)
n
< p < ˆp + z
α/2
r
ˆp(1 − ˆp)
n
.
✲ ❑❤♦↔♥❣ "✐♥ ❝➙② "E✐ ✤❛ ❝❤♦
p
✿
p < ˆp + z
α
r
ˆp(1 − ˆp)
n
.
✲ ❑❤♦↔♥❣ (✐♥ ❝➟② (-✐ (❤✐➸✉ ❝❤♦
p
✿
p > ˆp − z
α
r
ˆp(1 − ˆp)
n
.
❚!♥ ❚❤➜% ❚& ✷✾✴✹✵
❱➼ ❞# ✼
●✐❡♦ ✹✵✵ ❤↕( ✤➟✉ ①❛♥❤ (❤➻ ❝1 ✺✵ ❤↕( ❦❤4♥❣ ♥↔② ♠➛♠✳ ❱<✐ ✤= (✐♥ ❝➟② ✾✽✪✱ ❤➣② (➻♠✿
❛✮ ❑❤♦↔♥❣ (✐♥ ❝➟② ✤F✐ ①G♥❣ ❝❤♦ (➾ ❧➺ ❤↕( ♥↔② ♠➛♠✳
❜✮ ❑❤♦↔♥❣ (✐♥ ❝➟② (F✐ (❤✐➸✉ ❝❤♦ (➾ ❧➺ ❤↕( ❦❤4♥❣ ♥↔② ♠➛♠✳
❈❤♦ ❜✐➳(
z
0,05
= 1, 645; z
0,025
= 1, 96; z
0,02
= 2, 054; z
0,01
= 2, 326
✳
●✐↔✐✳
❛✳ ✲ ❚➾ ❧➺ ♠➝✉✿
ˆp =
k
n
=
400 − 50
400
= 0, 875
✳
✲ ✣9 (✐♥ ❝➟②✿
1 − α = 0, 98 ⇒ α = 0, 02; z
α/2
= z
0,01
= 2, 326
✲ ❙❛✐ ;-✿
ε = z
α/2
r
ˆp(1 − ˆp)
n
= 2, 326 ∗
r
0, 875 ∗ (1 − 0, 875)
400
= 0, 0385
✲ ❑❤♦↔♥❣ (✐♥ ❝➟② ❝❤♦ (➾ ❧➺ ❤↕( ♥↔② ♠➛♠ ♣✿
ˆp − ε < p < ˆp + ε ⇔ 0, 8365 < p < 0, 9135
❚!♥ ❚❤➜% ❚& ✸✵✴✹✵
❜✳ ✲ ❚➾ ❧➺ ♠➝✉✿
ˆp =
k
n
=
50
400
= 0, 125
✲ ✣, -✐♥ ❝➟②✿
1 − α = 0, 98 ⇒ α = 0, 02; z
α
= z
0,02
= 2, 054
✲ ❙❛✐ 56✿
ε = z
α
r
ˆp(1 − ˆp)
n
= 2, 054 ∗
r
0, 125 ∗ (1 − 0, 125)
400
= 0, 034
✲ ❑❤♦↔♥❣ -✐♥ ❝➟② -6✐ -❤✐➸✉ ❝❤♦ -➾ ❧➺ ❤↕- ❦❤?♥❣ ♥↔② ♠➛♠ ♣✿
p > ˆp − ε ⇔ p > 0, 091
❚!♥ ❚❤➜% ❚& ✸✶✴✹✵
❜✳ ◆❤➟♥ ①➨(
✐✮ ❑❤♦↔♥❣ (✐♥ ❝➟② ❝❤♦ 12 ❧45♥❣ ❝→ (❤➸
❚B ❦❤♦↔♥❣ -✐♥ ❝➟② ❝❤♦ -➾ ❧➺ -❛ ❝C -❤➸ 5✉② D❛ ❦❤♦↔♥❣ -✐♥ ❝➟② ❝❤♦ 56 ❧EF♥❣ ❝→ -❤➸ ♠❛♥❣
❞➜✉ ❤✐➺✉ ♥❣❤✐➯♥ ❝K✉ ❤♦➦❝ 56 ❧EF♥❣ ❝→ -❤➸ ❝M❛ -N♥❣ -❤➸✳ ❈P -❤➸✿
✲ ●R✐ ❑ ✈➔ ◆ ❧➛♥ ❧EF- ❧➔ 56 ❧EF♥❣ ❝→ -❤➸ ♠❛♥❣ ❞➜✉ ❤✐➺✉ ♥❣❤✐➯♥ ❝K✉ ✈➔ 56 ❧EF♥❣ ❝→ -❤➸
❝M❛ -N♥❣ -❤➸✳ ❑❤✐ ✤C✱ -➾ ❧➺ ❞➜✉ ❤✐➺✉ ♥❣❤✐➯♥ ❝K✉✿
p =
K
N
(1)
✲ ❉Y❛ -D➯♥ ♠➝✉ ✤✐➲✉ -D❛✱ -❛ E[❝ ❧EF♥❣ ✤EF❝ ❦❤♦↔♥❣ -✐♥ ❝➟② ❝❤♦ -➾ ❧➺ ♣✿
p
1
< p < p
2
(2)
❚B ✭✶✮ ✈➔ ✭✷✮ 5✉② D❛ ❦❤♦↔♥❣ -✐♥ ❝➟② ❝❤♦ ❣✐→ -D` ❑ ❤♦➦❝ ◆✳
❚!♥ ❚❤➜% ❚& ✸✷✴✹✵
❱➼ ❞; ✽
❚!♦♥❣ ♠&' ❝✉&❝ ❜➛✉ ❝, - ♠&' ✤/❛ ♣❤34♥❣ ❝5 ✶✵✵✵✵ ❝, '!✐✳ ◆❣3;✐ '❛ ♣❤<♥❣ ✈➜♥ ♥❣➝✉
♥❤✐➯♥ ✸✵✵ ❝, '!✐ '❤➻ '❤➜② ❝5 ✶✺✻ ♥❣3;✐ F♥❣ ❤& ❝❤♦ G♥❣ ❝, ✈✐➯♥ ❆✳ ❱J✐ ✤& '✐♥ ❝➟② ✾✺✪
'➻♠ ❦❤♦↔♥❣ '✐♥ ❝➟② ✭✤Q✐ ①G♥❣✮ ❝❤♦ TQ ❧3V♥❣ ❝, '!✐ - ✤/❛ ♣❤34♥❣ ♥➔② F♥❣ ❤& ❝❤♦ G♥❣ ❝,
✈✐➯♥ ❆✳
●✐↔✐✳
●R✐ ❑ ❧➔ 56 ❧EF♥❣ ❝a -D✐ b ✤`❛ ♣❤Ec♥❣ ♥➔② M♥❣ ❤, K♥❣ ❝a ✈✐➯♥ ❆✳ ❚❛ ❝C -➾ ❧➺ M♥❣
❤, K♥❣ ❝a ✈✐➯♥ ❆✿
p = K/10000
✳
❚➾ ❧➺ ♠➝✉✿
ˆp = k/n = 156/300 = 0, 52
✣, -✐♥ ❝➟②✿
1 − α = 0, 95 ⇒ α = 0, 05; z
α/2
= z
0,025
= 1, 96
❙❛✐ 56✿
ε = z
α/2
r
ˆp(1 − ˆp)
n
= 1, 96 ∗
r
0, 52 ∗ 0, 48
300
= 0, 0565
❑❤♦↔♥❣ -✐♥ ❝➟② ❝❤♦ -➾ ❧➺ M♥❣ ❤,✿
ˆp − ε < p < ˆp + ε ⇔ 0, 4635 < p < 0, 5765
❚B ✤C✱ 5✉② D❛✿
0, 4635 < K/10000 < 0, 5765 ⇔ 4635 < K < 5765
❚!♥ ❚❤➜% ❚& ✸✸✴✹✵
❱➼ ❞# ✾
✣➸ "#❝ ❧"&♥❣ )* ❧"&♥❣ ❝→ ❝, -.♦♥❣ ❤1✱ ♥❣"3✐ -❛ ❧➔♠ ♥❤" )❛✉✳ ❇➢- ♥❣➝✉ ♥❤✐➯♥ ✺✵✵ ❝♦♥
❝→✱ )❛✉ ✤, ✤→♥❤ ❞➜✉ ✈➔♦ ❝→❝ ❝♦♥ ❝→ ✤➣ ✤"&❝ ❜➢- ✈➔ -❤↔ ❝❤G♥❣ -❛ ①✉*♥❣ ❤1✳ ❙❛✉ ✤, ❜➢-
♥❣➝✉ ♥❤✐➯♥ ✷✵✵ ❝♦♥ ❝→ ✤➸ ❦✐➸♠ -.❛ -❤➻ -❤➜② ❝, ✸✵ ❝♦♥ ❝, ✤→♥❤ ❞➜✉✳ ❱#✐ ✤P -✐♥ ❝➟② ✾✵✪
-➻♠ ❦❤♦↔♥❣ -✐♥ ❝➟② ✤*✐ ①T♥❣ ❝❤♦ )* ❧"&♥❣ ❝→ ❝, -.♦♥❣ ❤1✳
●✐↔✐✳
●!✐ ◆ ❧➔ &' ❧()♥❣ ❝→ ./♦♥❣ ❤2✳ ❚❛ ❝6 .➾ ❧➺ ❝→ ✤()❝ ✤→♥❤ ❞➜✉✿
p = 500/N
✳
❚➾ ❧➺ ♠➝✉✿
ˆp = k/n = 30/200 = 0, 15
✣A .✐♥ ❝➟②✿
1 − α = 0, 9 ⇒ α = 0, 1; z
α/2
= z
0,05
= 1, 645
❙❛✐ &'✿
ε = z
α/2
r
ˆp(1 − ˆp)
n
= 1, 645 ∗
r
0, 15 ∗ 0, 85
200
= 0, 0415
❑❤♦↔♥❣ .✐♥ ❝➟② ❝❤♦ .➾ ❧➺ ❝→ ✤()❝ ✤→♥❤ ❞➜✉✿
ˆp − ε < p < ˆp + ε ⇔ 0, 1085 < p < 0, 1915
❚G ✤6✱ &✉② /❛✿
0, 1085 < 500/N < 0, 1915 ⇔ 2610, 9 < N < 4608, 3
❚!♥ ❚❤➜% ❚& ✸✹✴✹✵
✐✐✮ ❱➜♥ ✤➲ ✈➲ ❝0 ♠➝✉
✲ ❙❛✐ &' (J❝ ❧()♥❣✿
ε = z
α/2
r
ˆp(1 − ˆp)
n
✲ ✣✐➲✉ ❦✐➺♥ ❤↕♥ ❝❤➳✿
ε ≤ ∆
✱
∆ > 0
❝❤♦ ./(J❝✳
✲ ●✐↔ &O ❣✐→ ./P
ˆp
❦❤Q♥❣ ❜✐➳♥ ✤A♥❣ ❧J♥ ❦❤✐ .➼♥❤ .➼♥❤ ./➯♥ ❝→❝ ♠➝✉ ❦❤→❝ ♥❤❛✉✱ ❦❤✐ ✤6✿
n ≥
z
α/2
∆
2
ˆp(1 − ˆp).
✲ ❚/♦♥❣ ./(U♥❣ ❤)♣ ❦❤Q♥❣ ❝6 ❞W ❧✐➺✉ ✤➸ .➼♥❤ ❣✐→ ./P
ˆp
✱ &O ❞Y♥❣ ❜➜. ✤➥♥❣ .❤[❝
ˆp(1 − ˆp) ≤
1/4
.❛ .❤✉ ✤()❝ ❜➜. ✤➥♥❣ .❤[❝✿
n ≥
1
4
z
α/2
∆
2
.
❚!♥ ❚❤➜% ❚& ✸✺✴✹✵
❱➼ ❞# ✶✵
▼P- ❝✉P❝ ❦❤↔♦ )→- )T❝ ❦❤V❡ ❝P♥❣ ✤1♥❣ ✤❛♥❣ ✤"&❝ ❧➯♥ ❦➳ ❤♦↕❝❤ -.♦♥❣ ♠P- ❦❤✉ ✈Z❝ ✤[
-❤\ ❧#♥ ✈#✐ ♠]❝ ✤➼❝❤ "#❝ -➼♥❤ -✛ ❧➺ -.➫ ❡♠ -b ✵ ✤➳♥ ✶✹ -✉e✐ ❝❤"❛ ✤"&❝ -✐➯♠ ❝❤f♥❣ ❜↕✐
❧✐➺- ✤➛② ✤f✳ ❈→❝ ♥❤➔ -e ❝❤T❝ ❝f❛ ❞Z →♥ ♠✉*♥ -✛ ❧➺ ♠➝✉ ❝f❛ -.➫ ❦❤[♥❣ ✤"&❝ -✐➯♠ ❝❤f♥❣
✤➛② ✤f
k/n
♣❤↔✐ ♥➡♠ -.♦♥❣ ❦❤♦↔♥❣ ❝f❛ -✛ ❧➺ -❤Z❝
p
✈#✐ )❛✐ )*
±0, 05
✈#✐ ①→❝ )✉➜- -*✐
-❤✐➸✉ ✾✽✪✳ ❍V✐ ❦➼❝❤ -❤"#❝ ♠➝✉ ✤✐➲✉ -.❛ -*✐ -❤✐➸✉ ❧➔ ❜❛♦ ♥❤✐➯✉❄
❈❤♦ ❜✐➳-
z
0,05
= 1, 645; z
0,025
= 1, 96; z
0,02
= 2, 054; z
0,01
= 2, 326
✳
●✐↔✐✳
❙❛✐ &' (J❝ ❧()♥❣✿
ε = z
α/2
r
ˆp(1 − ˆp)
n
✣✐➲✉ ❦✐➺♥✿
ε ≤ ∆ = 0, 05
✳ ❙✉② /❛✿
n ≥
z
α/2
∆
2
ˆp(1 − ˆp).
❚!♥ ❚❤➜% ❚& ✸✻✴✹✵
❱➻
ˆp
❝❤$❛ ❜✐➳) ♥➯♥ →♣ ❞/♥❣ ❜➜) ✤➥♥❣ )❤4❝
ˆp(1 − ˆp) ≤ 1/4
)❛ )❤✉ ✤$6❝✿
n ≥
1
4
z
α/2
∆
2
.
✣9 )✐♥ ❝➟②✿
1 − α = 0, 98 ⇒ α = 0, 02; z
α/2
= z
0,01
= 2, 326
✳ ❉♦ ✤?✿
n ≥
1
4
2, 326
0, 05
2
= 541, 0276
❱➟②✱
n = 542
✳
❚!♥ ❚❤➜% ❚& ✸✼✴✹✵
❱➼ ❞# ✶✶
❙↔♥ ❧$%♥❣ ❣↕♦ ❜→♥ +❛ -+♦♥❣ ✶ ♥❣➔② ❝2 ♣❤➙♥ ♣❤6✐ ❝❤✉➞♥✳ ✣✐➲✉ -+❛ =↔♥ ❧$%♥❣ ❜→♥ +❛ -+♦♥❣
✶✷✵ ♥❣➔②✱ -❛ ✤$%❝ ❜↔♥❣ =6 ❧✐➺✉ =❛✉✿
❙↔♥ ❧$%♥❣ ✭❦❣✮ ✹✵✵✲✹✹✵ ✹✹✵✲✹✽✵ ✹✽✵✲✺✷✵ ✺✷✵✲✺✻✵ ✺✻✵✲✻✵✵ ✻✵✵✲✻✹✵ ✻✹✵✲✻✽✵
❙1 ♥❣➔② ✹ ✾ ✶✺ ✷✸ ✸✵ ✷✷ ✶✼
❛✮ ❱F✐ ✤G -✐♥ ❝➟② ✾✺✪ -➻♠ ❦❤♦↔♥❣ -✐♥ ❝➟② ✤6✐ ①P♥❣ ❝❤♦ =6 -✐➲♥ ❜→♥ +❛ -+✉♥❣ ❜➻♥❤ -+♦♥❣
♠G- ♥❣➔②✳ ❇✐➳- ❣✐→ ♠S✐ ❦❣ ❣↕♦ ❧➔ ✶✺✳✵✵✵ ✤T♥❣✳
❜✮ ◆❣➔② ❝2 =↔♥ ❧$%♥❣ ❣↕♦ ❜→♥ +❛ ❦❤V♥❣ ♥❤W ❤X♥ ✺✻✵ ❦❣ ✤$%❝ ❣Z✐ ❧➔ ♥❣➔② ❝❛♦ ✤✐➸♠✳ ❱F✐
✤G -✐♥ ❝➟② ✾✽✪✱ -➻♠ ❦❤♦↔♥❣ -✐♥ ❝➟② ✤6✐ ①P♥❣ ❝❤♦ =6 ♥❣➔② ❝❛♦ ✤✐➸♠ -+♦♥❣ ✶✵✵✵ ♥❣➔②
❜→♥✳
❈❤♦ ❜✐➳-
z
0,05
= 1, 645; z
0,025
= 1, 96; z
0,02
= 2, 054; z
0,01
= 2, 326
✳
❚!♥ ❚❤➜% ❚& ✸✽✴✹✵
●✐↔✐✳
❛✳ ❉↕♥❣ )❤✉ ❣B♥✿
❙↔♥ ❧$%♥❣ ✭❦❣✮ ✹✷✵ ✹✻✵ ✺✵✵ ✺✹✵ ✺✽✵ ✻✷✵ ✻✻✵
❙0 ♥❣➔② ✹ ✾ ✶✺ ✷✸ ✸✵ ✷✷ ✶✼
❈→❝ DE ✤➦❝ )G$♥❣ ♠➝✉✿
n = 120; ¯x = 566, 67; s = 64
✣9 )✐♥ ❝➟②✿
1 − α = 0, 95 ⇒ α = 0, 05; t
n−1;α/2
= t
119;0,025
≈ z
0,025
= 1, 96
✳
❙❛✐ DE✿
ε = t
n−1;α/2
s
√
n
= 1, 96 ∗
64
√
120
= 11, 45
❑❤♦↔♥❣ )✐♥ ❝➟② ❝❤♦ ❦❤E✐ ❧$6♥❣ ❣↕♦ )G✉♥❣ ❜➻♥❤ ❜→♥ ✤$6❝ )G♦♥❣ ✶ ♥❣➔②✿
¯x − ε < µ < ¯x + ε ⇔ 555, 22 < µ < 578, 12
❑❤♦↔♥❣ )✐♥ ❝➟② ❝❤♦ DE )✐➲♥ ❚ ✭♥❣➔♥ ✤T♥❣✮ )G✉♥❣ ❜➻♥❤ ❜→♥ ✤$6❝ )G♦♥❣ ✶ ♥❣➔②✿
555, 22 ∗ 15 < T < 578, 12 ∗ 15 ⇔ 8328, 3 < T < 8671, 8
❚!♥ ❚❤➜% ❚& ✸✾✴✹✵
❜✳ ●#✐ ❑ ❧➔ () ♥❣➔② ❝❛♦ ✤✐➸♠ 34♦♥❣ ✶✵✵✵ ♥❣➔②✳ ❚❛ ❝8 3➾ ❧➺ ♥❣➔② ❝❛♦ ✤✐➸♠✿
p = K/1000
✳
❚➾ ❧➺ ♠➝✉ ♥❣➔② ❝❛♦ ✤✐➸♠✿
ˆp =
k
n
=
30 + 22 + 17
120
= 0, 575
✣? 3✐♥ ❝➟②✿
1 − α = 0, 98 ⇒ α = 0, 02; z
α/2
= z
0,01
= 2, 326
✳
❙❛✐ ()✿
ε = z
α/2
r
ˆp(1 − ˆp)
n
= 2, 326 ∗
r
0, 575 ∗ (1 − 0, 575)
120
= 0, 105
❑❤♦↔♥❣ 3✐♥ ❝➟② ❝❤♦ 3➾ ❧➺ ♥❣➔② ❝❛♦ ✤✐➸♠ ♣✿
ˆp − ε < p < ˆp + ε ⇔ 0, 47 < p < 0, 68
❙✉② 4❛✿
0, 47 < K/1000 < 0, 68 ⇔ 470 < K < 680
❚!♥ ❚❤➜% ❚& ✹✵✴✹✵

Tài liệu khác của Đại học Duy Tân

Preview text:

❳⑩❈ ❙❯❻❚ ❚❍➮◆● ❑➊ ❚!♥ ❚❤➜% ❚&
✣➔ ◆➤♥❣✱ ✷✵✶✾ ❚!♥ ❚❤➜% ❚& ✶✴✹✵
❈❤"#♥❣ ✺✳ ()❝ ❧",♥❣ -❤❛♠ 01
✶✳ ,-❝ ❧01♥❣ ✤✐➸♠
✶✳✶ ✣6♥❤ ♥❣❤➽❛
✲ ❈❤♦ ♠➝✉ ♥❣➝✉ ♥❤✐➯♥ {X1, X2, ..., Xn} +, +-♥❣ +❤➸ ❝0 ♣❤➙♥ ♣❤3✐ ♣❤4 +❤✉5❝ ✈➔♦
+❤❛♠ 93 θ ❝❤:❛ ❜✐➳+✳ ❑❤✐ ✤0✱ ♠5+ ❤➔♠ ✭+❤3♥❣ ❦➯✮ ˆθ = ˆθ(X1, ..., Xn) ✤:D❝ ❣E✐ ❧➔ ♠5+
!❝ ❧ $♥❣ ❝G❛ +❤❛♠ 93 θ✳
✲ ❱I✐ ♠5+ ♠➝✉ ❣✐→ +KL ❝4 +❤➸ {x1, x2, ..., xn} +❛ +❤✉ ✤:D❝ ♠5+ ❣✐→ +KL ❝4 +❤➸ ❝G❛ ˆθ✳
❑❤✐ ✤0✱ ❣✐→ +KL ˆθ(x1, ..., xn) ✤:D❝ ❣E✐ ❧➔ !❝ ❧ $♥❣ ✤✐➸♠ ❝G❛ +❤❛♠ 93 θ ❞N❛ +K➯♥ ♠➝✉ ❣✐→ +KL {x1, x2, ..., xn}✳ ❚!♥ ❚❤➜% ❚& ✷✴✹✵
✶✳✷ :❤➙♥ ❧♦↕✐ 0-❝ ❧01♥❣
✲ OI❝ ❧:D♥❣ ˆθ = ˆθ(X1, ..., Xn) ✤:D❝ ❣E✐ ❧➔ !❝ ❧ $♥❣ ❦❤-♥❣ ❝❤➺❝❤ ❝G❛ +❤❛♠ 93 θ ♥➳✉ E(ˆ θ) = θ✳
❚K♦♥❣ +K:Q♥❣ ❤D♣ ♥❣:D❝ ❧↕✐ +❤➻ +❛ ❣E✐ ˆθ ✤:D❝ ❣E✐ ❧➔ !❝ ❧ $♥❣ ❝❤➺❝❤ ✈➔ ❣✐→ +KL b(θ) = E(ˆ
θ) − θ ✤:D❝ ❣E✐ ❧➔ ✤/ ❝❤➺❝❤ ❝0❛ !❝ ❧ $♥❣✳
✲ OI❝ ❧:D♥❣ ˆθ = ˆθ(X1, ..., Xn) ✤:D❝ ❣E✐ ❧➔ !❝ ❧ $♥❣ ❦❤-♥❣ ❝❤➺❝❤ 2✐➺♠ ❝➟♥ ❝G❛ +❤❛♠ 93 θ ♥➳✉ lim E(ˆ θ) = θ. n→+∞
✲ ❈❤♦ ˆθ1 ✈➔ ˆθ2 ❧➔ ❤❛✐ :I❝ ❧:D♥❣ ❦❤T♥❣ ❝❤➺❝❤ ❝G❛ +❤❛♠ 93 θ✳ ❚❛ ♥0✐ :I❝ ❧:D♥❣ ˆθ1 ❤✐➺✉
5✉↔ ❤7♥ :I❝ ❧:D♥❣ ˆθ2 ♥➳✉ D(ˆθ1) < D(ˆθ2)✳
✲ OI❝ ❧:D♥❣ ˆθ ❝G❛ θ ❧➔ :I❝ ❧:D♥❣ ❦❤T♥❣ ❝❤➺❝❤ ✈➔ ❝0 ♣❤:V♥❣ 9❛✐ D(ˆθ) ❜➨ ♥❤➜+ ✤:D❝
❣E✐ ❧➔ !❝ ❧ $♥❣ 282 ♥❤➜2✳ ❚!♥ ❚❤➜% ❚& ✸✴✹✵
!❝ ❧$%♥❣ ❦❤*♥❣ ❝❤➺❝❤ ❝,❛ ./✉♥❣ ❜➻♥❤ ✈➔ ♣❤$6♥❣ 7❛✐
✣:♥❤ ❧;✿ ❈❤♦ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥ X ❝,❛ ♠/0 01♥❣ 0❤➸ ❝3 E(X) = µ, D(X) = σ2✳ ●✐↔ 78
{X1, X2, ..., Xn} ❧➔ ♠➝✉ ♥❣➝✉ ♥❤✐➯♥ ❝,❛ X✳ ❑❤✐ ✤3✿ n ✐✮ 1 X = P X n
i ❧➔ ?@❝ ❧?A♥❣ ❦❤C♥❣ ❝❤➺❝❤ ❝,❛ µ✳ i=1 n ✐✐✮ 1 S2 = P (X n − 1
i − X)2 ❧➔ ?@❝ ❧?A♥❣ ❦❤C♥❣ ❝❤➺❝❤ ❝,❛ σ2✳ i=1
◆❤➟♥ ①➨.✿ ❈3 0❤➸ ❝❤E♥❣ ♠✐♥❤ ✤?A❝ F➡♥❣✿ n n − 1 1 X E(S2) = σ2 ✈@✐ S2 = (X ∗ n ∗ n i − ¯ X)2 i=1
✣✐➲✉ ♥➔② ❝3 ♥❣❤➽❛ S2 ❧➔ ?@❝ ❧?A♥❣ ❝❤➺❝❤ ✭❦❤C♥❣ ❝❤➺❝❤ 0✐➺♠ ❝➟♥✮ ❝,❛ σ2✱ ❞♦ ✤3 ♥3 ∗
✓❦❤C♥❣ ✤?A❝ ❝❤R♥✔ ❧➔♠ ♣❤?U♥❣ 7❛✐ ♠➝✉✳ ❚!♥ ❚❤➜% ❚& ✹✴✹✵
!❝ ❧$%♥❣ ❦❤*♥❣ ❝❤➺❝❤ ❝,❛ .➾ ❧➺
✣:♥❤ ❧;✿ ❈❤♦ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥ X ❝3 ♣❤➙♥ ♣❤W✐ ❇❡F♥♦✉❧❧✐ ✈@✐ 0❤❛♠ 7W p✳ ●R✐
(X1, X2, ..., Xn) ❧➔ ♠➝✉ ♥❣➝✉ ♥❤✐➯♥ ❝,❛ X✳ ❑❤✐ ✤3✿ ˆ X P = 1 + X2 + ... + Xn n
❧➔ ♠/0 ?@❝ ❧?A♥❣ ❦❤C♥❣ ❝❤➺❝❤ ❝,❛ 0❤❛♠ 7W p✳
B ♥❣❤➽❛✿ ◆➳✉ ❞➜✉ ❤✐➺✉ A ❝3 0➾ ❧➺ p ❝❤?❛ ❜✐➳0✱ ✈➔ 0F♦♥❣ ♠➝✉ ✤✐➲✉ 0F❛ ❦➼❝❤ 0❤?@❝ n ❝3 k
❝→ 0❤➸ ♠❛♥❣ ❞➜✉ ❤✐➺✉ A✱ 0❤➻ 0➛♥ 7✉➜0 fn = k/n ❧➔ ♠/0 ?@❝ ❧?A♥❣ ❦❤C♥❣ ❝❤➺❝❤ ❝,❛ p✳ ❚!♥ ❚❤➜% ❚& ✺✴✹✵ ❱➼ ❞G ✶
✲ ❈❤♦ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥ X ❝- ❦/ ✈1♥❣ µ ✈➔ ♣❤45♥❣ 6❛✐ σ2 ❝❤4❛ ❜✐➳8✳ ✣➸ 4<❝ ❧4>♥❣
❝❤♦ µ ✈➔ σ2 ♥❣4?✐ 8❛ 8✐➳♥ ❤➔♥❤ ✤✐➲✉ 8B❛ ♠➝✉ ❦➼❝❤ 8❤4<❝ ✷✵✵ ✈➔ 8➼♥❤ ✤4>❝ 8B✉♥❣ ❜➻♥❤ ✈➔
♣❤45♥❣ 6❛✐ ❝❤✉➞♥ ♠➝✉✿ ¯ x = 125, 8; s2 = 2, 76
❑❤✐ ✤-✱ 8❛ ❝- 8❤➸ ①❡♠ µ ≈ 125, 8 ✈➔ σ2 ≈ 2, 76✳
✲ ✣➸ 4<❝ ❧4>♥❣ 8➾ ❧➺ p ❝P 8B✐ Q♥❣ ❤R ❝❤♦ S♥❣ ❝P ✈✐➯♥ ❆✱ ♥❣4?✐ 8❛ ❦❤↔♦ 6→8 ♥❣➝✉ ♥❤✐➯♥
4000 ♥❣4?✐ 8❤➻ ❝- ✷✻✹✵ ♥❣4?✐ Q♥❣ ❤R S♥❣ ❝P ✈✐➯♥ ♥➔②✳ ◆❤4 ✈➟②✱ 8❛ ❝- 8❤➸ ①❡♠ 8➾ ❧➺ Q♥❣
❤R S♥❣ ❝P ✈✐➯♥ ❆ ①➜♣ ①➾✿ p ≈ 2640/4000 = 0, 66 ❚!♥ ❚❤➜% ❚& ✻✴✹✵ X θ {X1, X2, ..., Xn} L = L(X1, ..., Xn) U = U (X1, ..., Xn) α ∈ (0, 1)
P (L < θ < U ) = 1 − α. (L, U ) θ 1 − α α U − L (L, U ) θ 1 − α 1 − α α1, α2 ∈ (0, 1) α1 + α2 = α Gn = Gn(X1, ..., Xn, θ) Gn Gn n → ∞ θ Gn θ a b P (Gn ≤ a) = α1 P (Gn < b) = 1 − α2
a < Gn(X1, ..., Xn, θ) < b θ L, U
P (L < θ < U ) ≥ 1 − α. α1 = 0 α2 = 0
✷✳✷ ❑❤♦↔♥❣ (✐♥ ❝➟② ❝❤♦ ❦➻ ✈0♥❣ ❦❤✐ ✤➣ ❜✐➳( ♣❤67♥❣ 8❛✐
❇➔✐ (♦→♥✳ ❈❤♦ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥ X ❝-❛ ♠01 12♥❣ 1❤➸ ❝4 ♣❤➙♥ ♣❤7✐ ❝❤✉➞♥ N(µ; σ2) ✈:✐
µ ❝❤;❛ ❜✐➳1 ✈➔ σ2 ✤➣ ❜✐➳1✳ ❚➻♠ ❦❤♦↔♥❣ 1✐♥ ❝➟② ❝❤♦ µ ✈:✐ ♠E❝ F ♥❣❤➽❛ α✳
●>✐ ?✳ ◆➳✉ {X1, X2, ..., Xn} ❧➔ ♠➝✉ ♥❣➝✉ ♥❤✐➯♥ ❝-❛ X 1❤➻ X − µ Z = √ ∼ N(0; 1). σ/ n
✲ ❱:✐ α ∈ (0; 1)✱ ❝❤M♥ α1 = α2 = α/2 ✈➔ zα/2 = Φ−1(1 − α) ❧➔ ❣✐→ (@A (B✐ ❤↕♥ ♠E❝ 2
α/2 ❝-❛ ♣❤➙♥ ♣❤7✐ ❝❤✉➞♥ 1➢❝✱ ❣✐↔✐ ❜➜1 ♣❤;P♥❣ 1Q➻♥❤✿ σ σ
−zα/2 < Z < zα/2 ⇔ X − zα/2 √ < µ < X + z √ n α/2 n
1❛ ✤;S❝ ❦❤♦↔♥❣ 1✐♥ ❝➟② ✤7✐ ①E♥❣✳
✲ ◆➳✉ ❝❤M♥ α1 = 0 ❤♦➦❝ α2 = 0✳ ▲W❝ ♥➔② 1❛ 1❤❛② zα/ ❜X✐ 2
zα ✈➔ 1❤✉ ✤;S❝ ❦❤♦↔♥❣ 1✐♥
❝➟② ♠01 ♣❤➼❛✳ ❚!♥ ❚❤➜% ❚& ✶✵✴✹✵ ❚!♥ ❚❤➜% ❚& ✶✶✴✹✵ ❛✳ ❑➳( D✉↔
❱:✐ ✤0 1✐♥ ❝➟② 1 − α✿
✲ ❑❤♦↔♥❣ 1✐♥ ❝➟② ✤7✐ ①E♥❣ ❝-❛ µ✿ σ σ
x − zα/2 √ < µ < x + z √ , n α/2 n
✲ ❑❤♦↔♥❣ 1✐♥ ❝➟② 17✐ ✤❛ ❝-❛ µ✿ σ µ < x + zα √ . n
✲ ❑❤♦↔♥❣ 1✐♥ ❝➟② 17✐ 1❤✐➸✉ ❝-❛ µ✿ σ µ > x − zα √ . n ❚!♥ ❚❤➜% ❚& ✶✷✴✹✵ ❱➼ ❞# ✷
❑❤"✐ ❧%&♥❣ ✭❦❣✮ ❝-❛ ♠01 1❤✐➳1 ❜4 ❝5 ♣❤➙♥ ♣❤"✐ ❝❤✉➞♥ N(µ; σ2) ✈;✐ σ = 0.2 ✭❦❣✮✳ ❈❤>♥
♥❣➝✉ ♥❤✐➯♥ ✷✺ 1❤✐➳1 ❜4 ♥❣%C✐ 1❛ 1➼♥❤ ✤%&❝ 1F✉♥❣ ❜➻♥❤ ♠➝✉ x = 65, 1 ✭❦❣✮✳ ❱;✐ ✤0 1✐♥ ❝➟②
✾✺✪ ❤➣② 1➻♠ ❦❤♦↔♥❣ 1✐♥ ❝➟② ✭✤"✐ ①Q♥❣✮ ❝❤♦ ❦❤"✐ ❧%&♥❣ 1F✉♥❣ ❜➻♥❤ ❝-❛ 1❤✐➳1 ❜4 ♥➔②✳ ❈❤♦ ❜✐➳1 z0,025 = 1, 96✳ ●✐↔✐✳
✲ ❈→❝ $% ✤➦❝ ()*♥❣ ♠➝✉✿ n = 25; ¯x = 65, 1✳
✲ ✣3 (✐♥ ❝➟②✿ 1 − α = 0, 95 ⇒ α = 0, 05; zα/2 = z0,025 = 1, 96✳
✲ ❙❛✐ $% *9❝ ❧*;♥❣✿ σ 0, 2 ε = zα/2 √ = 1, 96 ∗ √ = 0, 0784 n 25
✲ ❑❤♦↔♥❣ (✐♥ ❝➟② ❝❤♦ ❦❤%✐ ❧*;♥❣ ()✉♥❣ ❜➻♥❤ µ ❝C❛ (❤✐➳( ❜E ♥➔②✿ ¯
x − ε < µ < ¯x + ε ⇔ 65, 02 < µ < 65, 18 ❚!♥ ❚❤➜% ❚& ✶✸✴✹✵
❜✳ ❱➜♥ ✤➲ ❝/ ♠➝✉
❚H ❝I♥❣ (❤J❝ ❦❤♦↔♥❣ (✐♥ ❝➟② ❝❤♦ µ (❛ (❤➜② )➡♥❣ $❛✐ $% ❝C❛ *9❝ ❧*;♥❣ |x − µ| ❜➨ ❤N♥ ❤♦➦❝ ❜➡♥❣ σ zα/ ✳ 2 √n
✲ ❙❛✐ $% *9❝ ❧*;♥❣✿ σ ε = zα/2 √n
✲ ✣✐➲✉ ❦✐➺♥ ❤↕♥ ❝❤➳✿ ε < ∆ ✈9✐ ∆ > 0 ❝❤♦ ()*9❝✳ σ ε = zα/2 √ < ∆ n
✲ ●✐↔✐ ❜➜( ♣❤*N♥❣ ()➻♥❤ (❤❡♦ n✿ z 2 α/ n > 2σ . ∆ ❚!♥ ❚❤➜% ❚& ✶✹✴✹✵ ❱➼ ❞# ✸
❚FT ❧↕✐ ✈;✐ ❱➼ ❞W ✷ ♥➳✉ ②➯✉ ❝➛✉ Y❛✐ Y" %;❝ ❧%&♥❣ ❦❤Z♥❣ ✈%&1 [✉→ ✵✱✵✺ 1❤➻ ✈;✐ ✤0 1✐♥ ❝➟②
✾✽✪ 1❛ ❝➛♥ ❝❤>♥ 1"✐ 1❤✐➸✉ ❜❛♦ ♥❤✐➯✉ 1❤✐➳1 ❜4 ✤➸ ❦❤↔♦ Y→1❄ ❈❤♦ ❜✐➳1 z0,01 = 2, 326✳ ●✐↔✐✳
✲ ❙❛✐ $% *9❝ ❧*;♥❣✿ σ ε = zα/2 √n
✲ ✣✐➲✉ ❦✐➺♥ ❤↕♥ ❝❤➳✿ ε < ∆ = 0, 05✳ ✲ ❙✉② )❛✿ z 2 α/ n > 2σ . ∆
✲ ✣3 (✐♥ ❝➟②✿ 1 − α = 0, 98 ⇒ α = 0, 2; zα/2 = z0,01 = 2, 326✳ ❉♦ ✤W✿ 2, 326 ∗ 0, 22 n > = 86, 56 0, 05
❱➟②✱ ❦➼❝❤ (❤*9❝ ♠➝✉ (%✐ (❤✐➸✉ ❝➛♥ ❦❤↔♦ $→(✿ n = 87 ❚!♥ ❚❤➜% ❚& ✶✺✴✹✵
✷✳✸ ❑❤♦↔♥❣ )✐♥ ❝➟② ❝❤♦ ❦➻ ✈1♥❣ ❦❤✐ ❝❤2❛ ❜✐➳) ♣❤27♥❣ 8❛✐
❇➔✐ )♦→♥✿ ❈❤♦ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥ X ❝-❛ ♠01 12♥❣ 1❤➸ ❝4 ♣❤➙♥ ♣❤7✐ ❝❤✉➞♥ N(µ; σ2) ✈:✐
µ ✈➔ σ2 ❝❤<❛ ❜✐➳1✳ ❚➻♠ ❦❤♦↔♥❣ 1✐♥ ❝➟② ❝❤♦ µ ✈:✐ ✤0 1✐♥ ❝➟② 1 − α✳
●=✐ >✳ ◆➳✉ {X1, X2, ..., Xn} ❧➔ ♠➝✉ ♥❣➝✉ ♥❤✐➯♥ ❝-❛ X 1❤➻ X − µ T = √ ∼ T S/ n n−1
✲ ❱:✐ ♠I❝ J ♥❣❤➽❛ α✱ ❝❤M♥ α1 = α2 = α/2 ✈➔ ❧➜② ❣✐→ 1PQ tn
❧➔ ❣✐→ )?@ )A✐ ❤↕♥ ❝-❛ −1;α/2
♣❤➙♥ ♣❤7✐ Tn−1✱ 1❛ ✤✲ ❈❤M♥ α1 = 0 ❤♦➦❝ α2 = 0✳ ▲V❝ ♥➔② 1❛ 1❤❛② tn ❜W✐ t −1;α/2
n−1;α ✈➔ 1❤✉ ✤1✐♥ ❝➟② ♠01 ♣❤➼❛✳ ❚!♥ ❚❤➜% ❚& ✶✻✴✹✵ ❚!♥ ❚❤➜% ❚& ✶✼✴✹✵ ❑➳) C✉↔
❱:✐ ✤0 1✐♥ ❝➟② 1 − α✿
✲ ❑❤♦↔♥❣ 1✐♥ ❝➟② ✤7✐ ①I♥❣ ❝-❛ µ✿ s s x − tn √ < µ < x + t √ , −1;α/2 n n−1;α/2 n
✲ ❑❤♦↔♥❣ 1✐♥ ❝➟② 17✐ ✤❛ ❝-❛ µ✿ s µ < x + tn . −1;α √n
✲ ❑❤♦↔♥❣ 1✐♥ ❝➟② 17✐ 1❤✐➸✉ ❝-❛ µ✿ s x − tn < µ. −1;α √n ❚!♥ ❚❤➜% ❚& ✶✽✴✹✵ ◆❤➟♥ ①➨&✿
✐✮ ❑❤✐ n > 30✿ tn ✳ −1;α/2 ≈ zα/2
✐✐✮ ❑❤✐ ❦➼❝❤ )❤*+❝ ♠➝✉ ❧+♥ ✭n > 30✮ )❤❡♦ ✤5♥❤ ❧6 ❣✐+✐ ❤↕♥ )9✉♥❣ )➙♠ ❦➳) <✉↔ )9➯♥ ✈➝♥
→♣ ❞C♥❣ ✤*D❝ ❞E )❤✐➳✉ ❣✐↔ )❤✐➳) ✈➲ ✤✐➲✉ ❦✐➺♥ ♣❤➙♥ ♣❤H✐ ❝❤✉➞♥ ❝J❛ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥ X✳ ❱➼ ❞+ ✹
▼!" ♠➝✉ ✶✻ ♣✐♥ ❞,♥❣ ❝❤♦ 1♠❛3"♣❤♦♥❡ ✤67❝ ❝❤8♥ ♥❣➝✉ ♥❤✐➯♥ ❝:❛ ❝;♥❣ "② ❆ ❝> "✉?✐ "❤8
"3✉♥❣ ❜➻♥❤ "➼♥❤ "3➯♥ ♠➝✉ x = 24308 ✭❣✐D✮ ✈➔ ✤! ❧➺❝❤ ❝❤✉➞♥ ♠➝✉ s = 727 ✭❣✐D✮✳ ●✐↔ 1N
3➡♥❣ "✉?✐ "❤8 ♣✐♥ 1♠❛3"♣❤♦♥❡ ❝> ♣❤➙♥ ♣❤Q✐ ❝❤✉➞♥✳ ❱S✐ ✤! "✐♥ ❝➟② ✾✺✪✱ ❤➣② "➻♠ ❦❤♦↔♥❣
"✐♥ ❝➟② ❝❤♦ "✉?✐ "❤8 "3✉♥❣ ❜➻♥❤ ❝:❛ ♣✐♥ 1♠❛3"♣❤♦♥❡ ✤67❝ 1↔♥ ①✉➜" ❜]✐ ❝;♥❣ "② ❆✳ ❈❤♦
❜✐➳" t15;0,025 = 2, 1314✳ ❚!♥ ❚❤➜% ❚& ✶✾✴✹✵
●✐↔✐✳ ❚❤❡♦ ❣✐↔ )❤✐➳)✿ n = 16; ¯ x = 24308; s = 727
✣O )✐♥ ❝➟②✿ 1 − α = 0, 95 ⇒ α = 0, 05; tn−1;α/2 = t15;0,025 = 2, 1314 ❙❛✐ SH✿ s 727 ε = tn √ = 2, 1314 ∗ √ = 387, 382 −1;α/2 n 16
❑❤♦↔♥❣ )✐♥ ❝➟② ❝❤♦ )✉T✐ )❤U )9✉♥❣ ❜➻♥❤ ❝J❛ ♣✐♥ S♠❛9)♣❤♦♥❡✿ ¯
x − ε < µ < ¯x + ε ⇔ 23920.618 < µ < 24695.382 ❚!♥ ❚❤➜% ❚& ✷✵✴✹✵ ❱➼ ❞+ ✺
◆➠♥❣ 1✉➜" ✭"↕✴❤❛✮ ❝:❛ ♠!" ❧♦↕✐ ❝➙② "3d♥❣ "✉➙♥ "❤❡♦ ❧✉➟" ♣❤➙♥ ♣❤Q✐ ❝❤✉➞♥✳ ❍➣② "➻♠ 6S❝
❧67♥❣ ❦❤♦↔♥❣ ✤Q✐ ①f♥❣ ♥➠♥❣ 1✉➜" "3✉♥❣ ❜➻♥❤ ❝:❛ ♠!" ❧♦↕✐ ❝➙② "3d♥❣ ♥➔② ✈S✐ ✤! "✐♥ ❝➟②
✾✺✪ "3➯♥ ❝g 1] ❜↔♥❣ 1Q ❧✐➺✉ ✤✐➲✉ "3❛ 1❛✉ ✤➙②✿
◆➠♥❣ 1✉➜"✭"↕✴❤❛✮
✹✷✲✹✼ ✹✼✲✺✷ ✺✷✲✺✼ ✺✼✲✻✷ ✻✷✲✻✼
❙Q ✤✐➸♠ "❤✉ ❤♦↕❝❤ ✷ ✺ ✶✹ ✶✵ ✺
❈❤♦ ❜✐➳" z0,05 = 1, 645; z0,025 = 1, 96; z0,02 = 2, 054; z0,01 = 2, 326✳
●✐↔✐✳ ❉↕♥❣ )❤✉ ❣U♥✿
◆➠♥❣ S✉➜)✭)↕✴❤❛✮
✹✹✱✺ ✹✾✱✺ ✺✹✱✺ ✺✾✱✺ ✻✹✱✺
❙H ✤✐➸♠ )❤✉ ❤♦↕❝❤ ✷ ✺ ✶✹ ✶✵ ✺ ❚!♥ ❚❤➜% ❚& ✷✶✴✹✵
❈→❝ #$ ✤➦❝ '()♥❣ ♠➝✉✿ n = 36; ¯ x = 56, 028; s = 5, 321
✣1 '✐♥ ❝➟②✿ 1 − α = 0, 95 ⇒ α = 0, 05; tn−1;α/2 ≈ zα/2 = z0,025 = 1, 96 ❙❛✐ #$✿ s 5, 321 ε = tn √ = 1, 96 ∗ √ = 1, 738 −1;α/2 n 36
❑❤♦↔♥❣ '✐♥ ❝➟② ❝❤♦ ♥➠♥❣ #✉➜' '(✉♥❣ ❜➻♥❤✿ ¯
x − ε < µ < ¯x + ε ⇔ 54, 29 < µ < 57, 766 ❚!♥ ❚❤➜% ❚& ✷✷✴✹✵
✷✳✹ ❑❤♦↔♥❣ )✐♥ ❝➟② ❝❤♦ ♣❤/0♥❣ 1❛✐ ✭✤5❝ )❤➯♠✮
❇➔✐ )♦→♥✿ ❈❤♦ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥ X ❝A❛ ♠1' 'B♥❣ '❤➸ ❝D ♣❤➙♥ ♣❤$✐ ❝❤✉➞♥ N(µ; σ2) ✈I✐
µ ✈➔ σ2 ❝❤)❛ ❜✐➳'✳ ❚➻♠ ❦❤♦↔♥❣ '✐♥ ❝➟② ❝❤♦ σ2 ✈I✐ ✤1 '✐♥ ❝➟② 1 − α✳
❑➳) =✉↔✿ ◆➳✉ {X1, X2, ..., Xn} ❧➔ ♠➝✉ ♥❣➝✉ ♥❤✐➯♥ ❝A❛ X '❤➻ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥✿ (n − 1)S2 χ2 = ∼ χ2 . σ2 n−1
✲ ❱I✐ ♠R❝ S ♥❣❤➽❛ α✱ ❝❤V♥ α1 = α2 = α/2 ✈➔ ❧➜② ❣✐→ '(W χ2 , χ2 ❧➔ ❣✐→ )@A n−1;α/2 n−1;1−α/2
)B✐ ❤↕♥ ♠R❝ α/2 ✈➔ 1 − α/2 ❝A❛ ♣❤➙♥ ♣❤$✐ χ2 ✱ '❛ ✤)X❝ ❦❤♦↔♥❣ '✐♥ ❝➟②✳ n−1
✲ ❈❤V♥ α1 = 0 ❤♦➦❝ α2 = 0✳ ▲Z❝ ♥➔② '❛ '❤❛② χ2 ❜[✐ χ2 ✱ χ2 ❜[✐ n−1;α/2 n−1;α n−1;1−α/2 χ2
✈➔ '❤✉ ✤)X❝ ❦❤♦↔♥❣ '✐♥ ❝➟② ♠1' ♣❤➼❛✳ n−1;1−α ❚!♥ ❚❤➜% ❚& ✷✸✴✹✵ ❚!♥ ❚❤➜% ❚& ✷✹✴✹✵ ❑➳" #✉↔
❱!✐ ✤$ %✐♥ ❝➟② 1 − α✿
✲ ❑❤♦↔♥❣ %✐♥ ❝➟② ❤❛✐ ♣❤➼❛ ❝❤♦ σ2✿ (n − 1)s2 (n − 1)s2 < σ2 < , χ2 χ2 n−1;α/2 n−1;1−α/2
✲ ❑❤♦↔♥❣ %✐♥ ❝➟② %4✐ ✤❛ ❝❤♦ ♣❤56♥❣ 7❛✐ σ2✿ (n − 1)s2 0 < σ2 < . χ2n−1;1−α
✲ ❑❤♦↔♥❣ %✐♥ ❝➟② %4✐ %❤✐➸✉ ❝❤♦ ♣❤56♥❣ 7❛✐ σ2✿
(n − 1)s2 < σ2 < +∞. χ2n−1;α ❚!♥ ❚❤➜% ❚& ✷✺✴✹✵ ❱➼ ❞) ✻
▼!" ❞➙② ❝❤✉②➲♥ "+ ✤!♥❣ ✤.♥❣ ❣↕♦ ✈➔♦ ❜❛♦ ✤56❝ ❦✐➸♠ ";❛✳ ❈➙♥ "❤> ✷✺ ❜❛♦ ❣↕♦ ❞♦ ❞➙②
❝❤✉②➲♥ ♥➔② "❤+❝ ❤✐➺♥ "❤➻ "➼♥❤ ✤56❝ ✤! ❧➺❝❤ ❝❤✉➞♥ ❦❤F✐ ❧56♥❣ ❣↕♦ ♠G✐ ❜❛♦ s = 0, 15
✭❦❣✮✳ ❱K✐ ✤! "✐♥ ❝➟② ✾✺✪ "➻♠ ❦❤♦↔♥❣ "✐♥ ❝➟② ❝❤♦ ♣❤5Q♥❣ R❛✐ ❝S❛ ❦❤F✐ ❧56♥❣ ❣↕♦ ✤56❝
✤.♥❣ ❝❤♦ ♠G✐ ❜❛♦ ❜T✐ ❞➙② ❝❤✉②➲♥ ♥➔②✳ ●✐↔ "❤✐➳" ❦❤F✐ ❧56♥❣ ❣↕♦ ✤56❝ ✤.♥❣ ❝❤♦ ♠G✐ ❜❛♦
"✉➙♥ "❤❡♦ ❧✉➟" ♣❤➙♥ ♣❤F✐ ❝❤✉➞♥✳ ❈❤♦ ❜✐➳"✿ χ2 = 39, 364✱ χ2 = 12, 401. 24;0.025 24;0.975
✣→♣ ./✿ 0, 0137 < σ2 < 0, 0435✳ ❚!♥ ❚❤➜% ❚& ✷✻✴✹✵
✷✳✺ ❑❤♦↔♥❣ "✐♥ ❝➟② ❝❤♦ "✛ ❧➺
❇➔✐ "♦→♥✿ ❳➨% ❞➜✉ ❤✐➺✉ A %@♦♥❣ %A♥❣ %❤➸ ❝B %➾ ❧➺ ❧➔ p ✭❝❤5❛ ❜✐➳%✮✳ ✣➸ ❦❤↔♦ 7→% ❣✐→ %@M p
%❛ %✐➳♥ ❤➔♥❤ ✤✐➲✉ %@❛ ♠$% ♠➝✉ ❦➼❝❤ %❤5!❝ n✳ ●✐↔ 7R %@♦♥❣ ♠➝✉ ♥➔② ❝B k ♣❤➛♥ %R ♠❛♥❣
❞➜✉ ❤✐➺✉ A✳ ❍➣② %➻♠ ❦❤♦↔♥❣ %✐♥ ❝➟② ❝❤♦ p✳ ❑➳" #✉↔✿
✲ ●W✐ X ❧➔ 74 ♣❤➛♥ %R ♠❛♥❣ ❞➜✉ ❤✐➺✉ A %@♦♥❣ ♠➝✉ ❦➼❝❤ %❤5!❝ n✱ %❛ ❝B X ∼ B(n, p)✳
❚❤❡♦ ✤M♥❤ ❧[ ❣✐!✐ ❤↕♥ %➼❝❤ ♣❤➙♥ ▼♦✐✈@❡✲▲❛♣❧❛❝❡✱ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥✿ X − np Z =
❝B ♣❤➙♥ ♣❤4✐ ①➜♣ ①➾ ❝❤✉➞♥ %➢❝ N(0, 1)✳ pnp(1 − p)
✲ ❱!✐ α ∈ (0; 1) ❝❤♦ %@5!❝✱ ❝❤W♥ α1 = α2 = α/2 %❛ ✤5e❝ ❦❤♦↔♥❣✿ r r p(1 − p) p(1 − p) ˆ p − zα/2 < p < ˆ p + z n α/2 n ✈!✐ ˆp = X/n = k/n✳ ❚!♥ ❚❤➜% ❚& ✷✼✴✹✵ ❚!♥ ❚❤➜% ❚& ✷✽✴✹✵
▼➦" ❦❤→❝✱ ❞♦ ˆp ❧➔ ♠-" ./❝ ❧.0♥❣ ❝3❛ p ♥➯♥ "❛ ❝6 "❤➸ ①❡♠ p ≈ ˆp ❦❤✐ n ❧/♥✳
✣✐➲✉ ❦✐➺♥✿ nˆp > 5 ✈➔ n(1 − ˆp) > 5✳ ❛✳ ❑➳, -✉↔
❱/✐ ✤- "✐♥ ❝➟② 1 − α✿
✲ ❑❤♦↔♥❣ "✐♥ ❝➟② ✤E✐ ①F♥❣ ❝❤♦ p✿ r r ˆ p(1 − ˆ p) ˆ p(1 − ˆ p) ˆ p − zα/2 < p < ˆ p + z . n α/2 n
✲ ❑❤♦↔♥❣ "✐♥ ❝➙② "E✐ ✤❛ ❝❤♦ p✿ r ˆp(1 − ˆp) p < ˆ p + zα . n
✲ ❑❤♦↔♥❣ (✐♥ ❝➟② (-✐ (❤✐➸✉ ❝❤♦ p✿ r ˆp(1 − ˆp) p > ˆ p − zα . n ❚!♥ ❚❤➜% ❚& ✷✾✴✹✵ ❱➼ ❞# ✼
●✐❡♦ ✹✵✵ ❤↕( ✤➟✉ ①❛♥❤ (❤➻ ❝1 ✺✵ ❤↕( ❦❤4♥❣ ♥↔② ♠➛♠✳ ❱<✐ ✤= (✐♥ ❝➟② ✾✽✪✱ ❤➣② (➻♠✿
❛✮ ❑❤♦↔♥❣ (✐♥ ❝➟② ✤F✐ ①G♥❣ ❝❤♦ (➾ ❧➺ ❤↕( ♥↔② ♠➛♠✳
❜✮ ❑❤♦↔♥❣ (✐♥ ❝➟② (F✐ (❤✐➸✉ ❝❤♦ (➾ ❧➺ ❤↕( ❦❤4♥❣ ♥↔② ♠➛♠✳
❈❤♦ ❜✐➳( z0,05 = 1, 645; z0,025 = 1, 96; z0,02 = 2, 054; z0,01 = 2, 326✳ ●✐↔✐✳
❛✳ ✲ ❚➾ ❧➺ ♠➝✉✿ k 400 − 50 ˆ p = = = 0, 875✳ n 400
✲ ✣9 (✐♥ ❝➟②✿ 1 − α = 0, 98 ⇒ α = 0, 02; zα/2 = z0,01 = 2, 326 r r ✲ ❙❛✐ ;-✿ ˆ p(1 − ˆ p) 0, 875 ∗ (1 − 0, 875) ε = zα/2 = 2, 326 ∗ = 0, 0385 n 400
✲ ❑❤♦↔♥❣ (✐♥ ❝➟② ❝❤♦ (➾ ❧➺ ❤↕( ♥↔② ♠➛♠ ♣✿ ˆ p − ε < p < ˆ
p + ε ⇔ 0, 8365 < p < 0, 9135 ❚!♥ ❚❤➜% ❚& ✸✵✴✹✵
❜✳ ✲ ❚➾ ❧➺ ♠➝✉✿ k 50 ˆ p = = = 0, 125 n 400
✲ ✣, -✐♥ ❝➟②✿ 1 − α = 0, 98 ⇒ α = 0, 02; zα = z0,02 = 2, 054 ✲ ❙❛✐ 56✿ r r ˆ p(1 − ˆ p) 0, 125 ∗ (1 − 0, 125) ε = zα = 2, 054 ∗ = 0, 034 n 400
✲ ❑❤♦↔♥❣ -✐♥ ❝➟② -6✐ -❤✐➸✉ ❝❤♦ -➾ ❧➺ ❤↕- ❦❤?♥❣ ♥↔② ♠➛♠ ♣✿ p > ˆ p − ε ⇔ p > 0, 091 ❚!♥ ❚❤➜% ❚& ✸✶✴✹✵ ❜✳ ◆❤➟♥ ①➨(
✐✮ ❑❤♦↔♥❣ (✐♥ ❝➟② ❝❤♦ 12 ❧45♥❣ ❝→ (❤➸
❚B ❦❤♦↔♥❣ -✐♥ ❝➟② ❝❤♦ -➾ ❧➺ -❛ ❝C -❤➸ 5✉② D❛ ❦❤♦↔♥❣ -✐♥ ❝➟② ❝❤♦ 56 ❧EF♥❣ ❝→ -❤➸ ♠❛♥❣
❞➜✉ ❤✐➺✉ ♥❣❤✐➯♥ ❝K✉ ❤♦➦❝ 56 ❧EF♥❣ ❝→ -❤➸ ❝M❛ -N♥❣ -❤➸✳ ❈P -❤➸✿
✲ ●R✐ ❑ ✈➔ ◆ ❧➛♥ ❧EF- ❧➔ 56 ❧EF♥❣ ❝→ -❤➸ ♠❛♥❣ ❞➜✉ ❤✐➺✉ ♥❣❤✐➯♥ ❝K✉ ✈➔ 56 ❧EF♥❣ ❝→ -❤➸
❝M❛ -N♥❣ -❤➸✳ ❑❤✐ ✤C✱ -➾ ❧➺ ❞➜✉ ❤✐➺✉ ♥❣❤✐➯♥ ❝K✉✿ K p = (1) N
✲ ❉Y❛ -D➯♥ ♠➝✉ ✤✐➲✉ -D❛✱ -❛ E[❝ ❧EF♥❣ ✤EF❝ ❦❤♦↔♥❣ -✐♥ ❝➟② ❝❤♦ -➾ ❧➺ ♣✿ p1 < p < p2 (2)
❚B ✭✶✮ ✈➔ ✭✷✮ 5✉② D❛ ❦❤♦↔♥❣ -✐♥ ❝➟② ❝❤♦ ❣✐→ -D` ❑ ❤♦➦❝ ◆✳ ❚!♥ ❚❤➜% ❚& ✸✷✴✹✵ ❱➼ ❞; ✽
❚!♦♥❣ ♠&' ❝✉&❝ ❜➛✉ ❝, - ♠&' ✤/❛ ♣❤34♥❣ ❝5 ✶✵✵✵✵ ❝, '!✐✳ ◆❣3;✐ '❛ ♣❤<♥❣ ✈➜♥ ♥❣➝✉
♥❤✐➯♥ ✸✵✵ ❝, '!✐ '❤➻ '❤➜② ❝5 ✶✺✻ ♥❣3;✐ F♥❣ ❤& ❝❤♦ G♥❣ ❝, ✈✐➯♥ ❆✳ ❱J✐ ✤& '✐♥ ❝➟② ✾✺✪
'➻♠ ❦❤♦↔♥❣ '✐♥ ❝➟② ✭✤Q✐ ①G♥❣✮ ❝❤♦ TQ ❧3V♥❣ ❝, '!✐ - ✤/❛ ♣❤34♥❣ ♥➔② F♥❣ ❤& ❝❤♦ G♥❣ ❝, ✈✐➯♥ ❆✳
●✐↔✐✳ ●R✐ ❑ ❧➔ 56 ❧EF♥❣ ❝a -D✐ b ✤`❛ ♣❤Ec♥❣ ♥➔② M♥❣ ❤, K♥❣ ❝a ✈✐➯♥ ❆✳ ❚❛ ❝C -➾ ❧➺ M♥❣
❤, K♥❣ ❝a ✈✐➯♥ ❆✿ p = K/10000✳
❚➾ ❧➺ ♠➝✉✿ ˆp = k/n = 156/300 = 0, 52
✣, -✐♥ ❝➟②✿ 1 − α = 0, 95 ⇒ α = 0, 05; zα/2 = z0,025 = 1, 96 ❙❛✐ 56✿ r ˆp(1 − ˆp) r 0, 52 ∗ 0, 48 ε = zα/2 = 1, 96 ∗ = 0, 0565 n 300
❑❤♦↔♥❣ -✐♥ ❝➟② ❝❤♦ -➾ ❧➺ M♥❣ ❤,✿ ˆ p − ε < p < ˆ
p + ε ⇔ 0, 4635 < p < 0, 5765 ❚B ✤C✱ 5✉② D❛✿
0, 4635 < K/10000 < 0, 5765 ⇔ 4635 < K < 5765 ❚!♥ ❚❤➜% ❚& ✸✸✴✹✵ ❱➼ ❞# ✾
✣➸ "#❝ ❧"&♥❣ )* ❧"&♥❣ ❝→ ❝, -.♦♥❣ ❤1✱ ♥❣"3✐ -❛ ❧➔♠ ♥❤" )❛✉✳ ❇➢- ♥❣➝✉ ♥❤✐➯♥ ✺✵✵ ❝♦♥
❝→✱ )❛✉ ✤, ✤→♥❤ ❞➜✉ ✈➔♦ ❝→❝ ❝♦♥ ❝→ ✤➣ ✤"&❝ ❜➢- ✈➔ -❤↔ ❝❤G♥❣ -❛ ①✉*♥❣ ❤1✳ ❙❛✉ ✤, ❜➢-
♥❣➝✉ ♥❤✐➯♥ ✷✵✵ ❝♦♥ ❝→ ✤➸ ❦✐➸♠ -.❛ -❤➻ -❤➜② ❝, ✸✵ ❝♦♥ ❝, ✤→♥❤ ❞➜✉✳ ❱#✐ ✤P -✐♥ ❝➟② ✾✵✪
-➻♠ ❦❤♦↔♥❣ -✐♥ ❝➟② ✤*✐ ①T♥❣ ❝❤♦ )* ❧"&♥❣ ❝→ ❝, -.♦♥❣ ❤1✳
●✐↔✐✳ ●!✐ ◆ ❧➔ &' ❧()♥❣ ❝→ ./♦♥❣ ❤2✳ ❚❛ ❝6 .➾ ❧➺ ❝→ ✤()❝ ✤→♥❤ ❞➜✉✿ p = 500/N✳
❚➾ ❧➺ ♠➝✉✿ ˆp = k/n = 30/200 = 0, 15
✣A .✐♥ ❝➟②✿ 1 − α = 0, 9 ⇒ α = 0, 1; zα/2 = z0,05 = 1, 645 ❙❛✐ &'✿ r ˆp(1 − ˆp) r 0, 15 ∗ 0, 85 ε = zα/2 = 1, 645 ∗ = 0, 0415 n 200
❑❤♦↔♥❣ .✐♥ ❝➟② ❝❤♦ .➾ ❧➺ ❝→ ✤()❝ ✤→♥❤ ❞➜✉✿ ˆ p − ε < p < ˆ
p + ε ⇔ 0, 1085 < p < 0, 1915
❚G ✤6✱ &✉② /❛✿
0, 1085 < 500/N < 0, 1915 ⇔ 2610, 9 < N < 4608, 3 ❚!♥ ❚❤➜% ❚& ✸✹✴✹✵
✐✐✮ ❱➜♥ ✤➲ ✈➲ ❝0 ♠➝✉
✲ ❙❛✐ &' (J❝ ❧()♥❣✿ r ˆp(1 − ˆp) ε = zα/2 n
✲ ✣✐➲✉ ❦✐➺♥ ❤↕♥ ❝❤➳✿ ε ≤ ∆✱ ∆ > 0 ❝❤♦ ./(J❝✳
✲ ●✐↔ &O ❣✐→ ./P ˆp ❦❤Q♥❣ ❜✐➳♥ ✤A♥❣ ❧J♥ ❦❤✐ .➼♥❤ .➼♥❤ ./➯♥ ❝→❝ ♠➝✉ ❦❤→❝ ♥❤❛✉✱ ❦❤✐ ✤6✿ z 2 α/ n ≥ 2 ˆ p(1 − ˆ p). ∆
✲ ❚/♦♥❣ ./(U♥❣ ❤)♣ ❦❤Q♥❣ ❝6 ❞W ❧✐➺✉ ✤➸ .➼♥❤ ❣✐→ ./P ˆp✱ &O ❞Y♥❣ ❜➜. ✤➥♥❣ .❤[❝ ˆp(1 − ˆp) ≤
1/4 .❛ .❤✉ ✤()❝ ❜➜. ✤➥♥❣ .❤[❝✿ 1 z 2 α/ n ≥ 2 . 4 ∆ ❚!♥ ❚❤➜% ❚& ✸✺✴✹✵ ❱➼ ❞# ✶✵
▼P- ❝✉P❝ ❦❤↔♦ )→- )T❝ ❦❤V❡ ❝P♥❣ ✤1♥❣ ✤❛♥❣ ✤"&❝ ❧➯♥ ❦➳ ❤♦↕❝❤ -.♦♥❣ ♠P- ❦❤✉ ✈Z❝ ✤[
-❤\ ❧#♥ ✈#✐ ♠]❝ ✤➼❝❤ "#❝ -➼♥❤ -✛ ❧➺ -.➫ ❡♠ -b ✵ ✤➳♥ ✶✹ -✉e✐ ❝❤"❛ ✤"&❝ -✐➯♠ ❝❤f♥❣ ❜↕✐
❧✐➺- ✤➛② ✤f✳ ❈→❝ ♥❤➔ -e ❝❤T❝ ❝f❛ ❞Z →♥ ♠✉*♥ -✛ ❧➺ ♠➝✉ ❝f❛ -.➫ ❦❤[♥❣ ✤"&❝ -✐➯♠ ❝❤f♥❣
✤➛② ✤f k/n ♣❤↔✐ ♥➡♠ -.♦♥❣ ❦❤♦↔♥❣ ❝f❛ -✛ ❧➺ -❤Z❝ p ✈#✐ )❛✐ )* ±0, 05 ✈#✐ ①→❝ )✉➜- -*✐
-❤✐➸✉ ✾✽✪✳ ❍V✐ ❦➼❝❤ -❤"#❝ ♠➝✉ ✤✐➲✉ -.❛ -*✐ -❤✐➸✉ ❧➔ ❜❛♦ ♥❤✐➯✉❄
❈❤♦ ❜✐➳- z0,05 = 1, 645; z0,025 = 1, 96; z0,02 = 2, 054; z0,01 = 2, 326✳
●✐↔✐✳ ❙❛✐ &' (J❝ ❧()♥❣✿ r ˆp(1 − ˆp) ε = zα/2 n
✣✐➲✉ ❦✐➺♥✿ ε ≤ ∆ = 0, 05✳ ❙✉② /❛✿ z 2 α/ n ≥ 2 ˆ p(1 − ˆ p). ∆ ❚!♥ ❚❤➜% ❚& ✸✻✴✹✵
❱➻ ˆp ❝❤$❛ ❜✐➳) ♥➯♥ →♣ ❞/♥❣ ❜➜) ✤➥♥❣ )❤4❝ ˆp(1 − ˆp) ≤ 1/4 )❛ )❤✉ ✤$6❝✿ 1 z 2 α/ n ≥ 2 . 4 ∆
✣9 )✐♥ ❝➟②✿ 1 − α = 0, 98 ⇒ α = 0, 02; zα/2 = z0,01 = 2, 326✳ ❉♦ ✤?✿ 1 2, 326 2 n ≥ = 541, 0276 4 0, 05 ❱➟②✱ n = 542✳ ❚!♥ ❚❤➜% ❚& ✸✼✴✹✵ ❱➼ ❞# ✶✶
❙↔♥ ❧$%♥❣ ❣↕♦ ❜→♥ +❛ -+♦♥❣ ✶ ♥❣➔② ❝2 ♣❤➙♥ ♣❤6✐ ❝❤✉➞♥✳ ✣✐➲✉ -+❛ =↔♥ ❧$%♥❣ ❜→♥ +❛ -+♦♥❣
✶✷✵ ♥❣➔②✱ -❛ ✤$%❝ ❜↔♥❣ =6 ❧✐➺✉ =❛✉✿
❙↔♥ ❧$%♥❣ ✭❦❣✮ ✹✵✵✲✹✹✵ ✹✹✵✲✹✽✵ ✹✽✵✲✺✷✵ ✺✷✵✲✺✻✵ ✺✻✵✲✻✵✵ ✻✵✵✲✻✹✵ ✻✹✵✲✻✽✵ ❙1 ♥❣➔② ✹ ✾ ✶✺ ✷✸ ✸✵ ✷✷ ✶✼
❛✮ ❱F✐ ✤G -✐♥ ❝➟② ✾✺✪ -➻♠ ❦❤♦↔♥❣ -✐♥ ❝➟② ✤6✐ ①P♥❣ ❝❤♦ =6 -✐➲♥ ❜→♥ +❛ -+✉♥❣ ❜➻♥❤ -+♦♥❣
♠G- ♥❣➔②✳ ❇✐➳- ❣✐→ ♠S✐ ❦❣ ❣↕♦ ❧➔ ✶✺✳✵✵✵ ✤T♥❣✳
❜✮ ◆❣➔② ❝2 =↔♥ ❧$%♥❣ ❣↕♦ ❜→♥ +❛ ❦❤V♥❣ ♥❤W ❤X♥ ✺✻✵ ❦❣ ✤$%❝ ❣Z✐ ❧➔ ♥❣➔② ❝❛♦ ✤✐➸♠✳ ❱F✐
✤G -✐♥ ❝➟② ✾✽✪✱ -➻♠ ❦❤♦↔♥❣ -✐♥ ❝➟② ✤6✐ ①P♥❣ ❝❤♦ =6 ♥❣➔② ❝❛♦ ✤✐➸♠ -+♦♥❣ ✶✵✵✵ ♥❣➔② ❜→♥✳
❈❤♦ ❜✐➳- z0,05 = 1, 645; z0,025 = 1, 96; z0,02 = 2, 054; z0,01 = 2, 326✳ ❚!♥ ❚❤➜% ❚& ✸✽✴✹✵ ●✐↔✐✳
❛✳ ❉↕♥❣ )❤✉ ❣B♥✿
❙↔♥ ❧$%♥❣ ✭❦❣✮ ✹✷✵ ✹✻✵ ✺✵✵ ✺✹✵ ✺✽✵ ✻✷✵ ✻✻✵ ❙0 ♥❣➔② ✹ ✾ ✶✺ ✷✸ ✸✵ ✷✷ ✶✼
❈→❝ DE ✤➦❝ )G$♥❣ ♠➝✉✿ n = 120; ¯x = 566, 67; s = 64
✣9 )✐♥ ❝➟②✿ 1 − α = 0, 95 ⇒ α = 0, 05; tn−1;α/2 = t119;0,025 ≈ z0,025 = 1, 96✳ ❙❛✐ DE✿ s 64 ε = tn √ = 1, 96 ∗ √ = 11, 45 −1;α/2 n 120
❑❤♦↔♥❣ )✐♥ ❝➟② ❝❤♦ ❦❤E✐ ❧$6♥❣ ❣↕♦ )G✉♥❣ ❜➻♥❤ ❜→♥ ✤$6❝ )G♦♥❣ ✶ ♥❣➔②✿ ¯
x − ε < µ < ¯x + ε ⇔ 555, 22 < µ < 578, 12
❑❤♦↔♥❣ )✐♥ ❝➟② ❝❤♦ DE )✐➲♥ ❚ ✭♥❣➔♥ ✤T♥❣✮ )G✉♥❣ ❜➻♥❤ ❜→♥ ✤$6❝ )G♦♥❣ ✶ ♥❣➔②✿
555, 22 ∗ 15 < T < 578, 12 ∗ 15 ⇔ 8328, 3 < T < 8671, 8 ❚!♥ ❚❤➜% ❚& ✸✾✴✹✵
❜✳ ●#✐ ❑ ❧➔ () ♥❣➔② ❝❛♦ ✤✐➸♠ 34♦♥❣ ✶✵✵✵ ♥❣➔②✳ ❚❛ ❝8 3➾ ❧➺ ♥❣➔② ❝❛♦ ✤✐➸♠✿ p = K/1000✳
❚➾ ❧➺ ♠➝✉ ♥❣➔② ❝❛♦ ✤✐➸♠✿ k 30 + 22 + 17 ˆ p = = = 0, 575 n 120
✣? 3✐♥ ❝➟②✿ 1 − α = 0, 98 ⇒ α = 0, 02; zα/2 = z0,01 = 2, 326✳ ❙❛✐ ()✿ r r ˆ p(1 − ˆ p) 0, 575 ∗ (1 − 0, 575) ε = zα/2 = 2, 326 ∗ = 0, 105 n 120
❑❤♦↔♥❣ 3✐♥ ❝➟② ❝❤♦ 3➾ ❧➺ ♥❣➔② ❝❛♦ ✤✐➸♠ ♣✿ ˆ p − ε < p < ˆ
p + ε ⇔ 0, 47 < p < 0, 68 ❙✉② 4❛✿
0, 47 < K/1000 < 0, 68 ⇔ 470 < K < 680 ❚!♥ ❚❤➜% ❚& ✹✵✴✹✵

Tài liệu liên quan: