11 -

11 -

20041

11 -

1.

2.

20042

3.

4.

5.

11 -

20043

t

t

Excel

11 - 11.1 Concept30-40 n=36,

Q: 903.92 ? ? ?

92.903X

20044

?, ()

,

11 - E X: , 67, 836.92 ~ 970.92: [30~40836.92 ~ 970.92]: [30~40()836.92 ~

970.9295%]836.92: lower limit of the interval970.92: upper limit of the interval

Q:

92.903X

20045

Q: Depend on1. 2. (confidence level)

X X

11 -

(

20046

)

Note: ,

11 - ()()

Note:1-,. 95%95%

20047

: 90%, 95%, 99%

11 -

11.1

)(xf

20048

92.903X x92.97092.836

11 - 11.2

. (n>=30),(),

11.2.1

X

2

20049

EX: ,144n=144 =576

Q: ?

2

11 -

1

2

200410

3

4

11 - 1. , ,

EX: 14480,

2. =, ,, , ,

X

80XX

X X nX 22 nX

200411

, ,

1.96 0.95 1.96 0.951.96 =1.96*2=3.92 ()

80X 5762 214424 nX

XX X

X

11 - 11.2 X

)(xf

2144

24 n

x

200412

x80x

11 -

11.3 1.96 X

f (x )_

200413

2.5% 2.5%

x

z0-1.96 1.96

_

3.923.92

95% X

11 -

3. ,0.95,

95%

95.0)96.1( XXP

95.0)96.196.1( XX XP

95.0)96.196.1( XX XXP

)96.1(~)96.1( XX XX XX 96.1

200414

note:,,,,. , .

)96.1(~)96.1( XX XX X

11 - 95% (important !!), (11.6)

, 95%,5%

a. 95%11.4,X

200415

a. 95%11.4,,

b. 5%,(11.6)

65321 ,,,, XXXXX

4X

11 - 11.4

)(xf

X95%

2x92.3296.12 xZ

200416

92.3

x92.3

92.31x2x

92.32 x3x

92.33 x4x

92.34 x5x

92.35 x6x 92.36 x

1x

11 -

95%

95%

95%

200417

0.95 0.05

11 - 4. (11.6),

X

XX 96.180X 2X

92.380296.1 X

92.8308.76

200418

76.08~83.9295%, 11.5

: 95%,76.08 ~ 83.92

11 -

f x( )

11.5

200419

xX X 196 83 92. .X X 196 76 08. .

X 80

11 -

1-,

X XZ 2 1)( 2 XZXP

1)( 22 XX ZXZP

200420

=10%, =5% , =1%,

1-,

645.12 Z 96.12 Z 575.22 Z

11 - 11.6 1 X

f x( )

( )1 X

200421

x

/ 2

( )1 X

Z X/2 Z X/2

/ 2

11 - 1-1

1 X

X Z X /2

200422

X Z X /2

XZX 2/ XZX 2/

11 -

11.1

1 2/ 2/Z XZX 2/

090. 010. 005. 1.645 XX 645.1

200423

095. 005. 0025. 1.96 XX 96.1

099. 001. 0005. 2.575 XX 575.2

11 -

11.7

200424

11 - EX11.1 .

100, , . , 8, :1.?2. 95%, ?3. 95%, ?

Solution:

2.21X

200425

Solution:,95%, ,=8, n=100,:

1. 2. ,

. 95%, 1.96

2.21X

XX ),(~ 2XN

95.0)96.1( XXP X

11 -

95%,1.568.

3. 1-,(11.8)

8.01008

nX

568.18.096.1 X

2ZX

768.22~632.19)568.12.21(~)568.12.21(8.096.12.21 ZX

200426

19.632 ~ 22.768 95%

; 19.632 ~ 22.768,()95%.

768.22~632.19)568.12.21(~)568.12.21(8.096.12.212 ZX768.22632.19

11 -

11.8

)(xf

200427

x

632.192.21X

768.22

11 -

X nX / n X

200428

X nX / n X

11 - EX11.2 ,1. 90%,?95%??

2. 95%,400,?100??

Solution:1. 645.105.02 ZZ

200429

(11.8)95%

2. n=400,

100

05.02

516.22884.198.0645.12.218.0645.12.21

8.0645.12.212

ZX

4.04008

nX

984.21416.204.0645.12.214.0645.12.21

4.0645.12.212

ZX

11 -

X Z Sn /2

nSZX / nSZX /

11.2.2

200430

nSZX /2/ nSZX /2/

11 - EX11.3 ,

95%Solution: n=36, , s=205, (11.9)

92.903X

97.6692.90336

20596.192.9032

n

SZX

200431

,(),95%,

97.6692.903 89.97095.836

89.97095.836

11 -

11.2

200432

11 - 11.3

,(n

11 - EX11.4 25 ,

S=696. 95%? (=720)

Solution:,=720. n=25, ,(11.10)

810,2X

810,2X

2822810720

96.12810 ZX

200434

95%,2,5383,092

282281025

72096.128102

nZX

092,3528,2

11 - 11.3.2

. ,

n-1t,,,t

t

2

1

)( 2

n

XXS

nSX

200435

t, ,

, S t

X ),(~2

nNX

)1,0(~ Nn

XZ

1~

ntnS

X

11 - 1-1

X t Snn 1 2, /

200436

n

1n t

11 - t

, ,Uv,,

vt

),(~ 2NX 2~ vU

v

Z

v

X

U

X

vv22

)()(

200437

vt

)( X

vv2

11 -

tt0vt

tt1

t

200438

tt1

tvt

( v ) t )1,0(~ Nt

v 30v t

11 -

f (t )

3 30

:1.

11.9 t0

200439

-5 -4 -3 -2 -1 0 1 2 3 4 5t

11 -

2. t,t,

11.10 t

200440

-5 -4 -3 -2 -1 0 1 2 3 4 5t z

t3

11 - 3. t

E(t)=0, V(t)= v / v-2V(t)= v / v-2 > 1. v (v >30), t1

v

11 - (degree of freedom)

(). n-1

S2 , ,(X1,X2,,Xn)

1

)(1

2

2

n

XXS

n

ii

X X n

XX

n

ii 1

200442

S , ,(X1,X2,,Xn)n , n-1(X1,X2,,Xn-1),Xn ,S2n-1

X nX

11 - T11.3 t

d.f. t0.10 t0.05 t0.025 t0.010 t0.005 d.f.1 3.078 6.314 12.706 31.821 63.657 12 1.886 2.920 4.303 6.965 10.925 23 1.638 2.353 3.182 4.541 5.841 34 1.533 2.132 2.776 3.747 4.604 4

200443

4 1.533 2.132 2.776 3.747 4.604 45 1.473 2.015 2.571 3.365 4.032 56 1.440 1.943 2.447 3.143 3.707 6 27 1.314 1.703 2.052 2.473 2.771 2728 1.313 1.701 2.048 2.467 2.763 2829 1.311 1.699 2.045 2.462 2.756 29

1.282 1.645 1.960 2.326 2.575

11 -

f (t )

11.11 t

200444

=0.025v2

0 4.303 t

11 -

f (t )

11.12 t

200445

v =5

2.015_2.015 t0

0.050.05

11 -

11.13 t

200446

11 -

11.14 t

200447

11 - t

t 1-,

1-,

nSX

2,12,1 , nn tt

Xn

St n 2,1

12,1

nS

tXP n

200448

11.15(1-)

1)( 2,12,1n

StX

nS

tXP nn

11 - 11.15 ( )1 X

f x( )

200449

t s nn 1 2, / / t s nn 1 2, / / x

1

0 tn1 2, / tn 1 2, / t XS n

/

11 -

X t Snn 1 2, /

200450

n

1n t

11 - 11.15 ,

. 16, 46,5.7.? (95%)?

Solution:n=16,. (11.13)

200451

t15,0.025=2.131,

95%,42.9649.04

037.346425.1131.246167.5

46 025.0,1162,1 tn

StX n

037.49963.42

11 - EX11.6 ,,,,

?

Solution:

79.246167.5

46 025..02 Zn

SZX

79.4821.43

200452

Ex11.15t(42.96~49.07).

, (43.21 ~48.79)95%.

79.4821.43

11 - 11.3.3 ,

, ,

. ,

EX:90%,

1-(1/K2),

2

X

),(~2

nX

2

)( XXK

200453

1-(1/K ), XK

2

11)(

KKXP X

11 - EX11.7

.200,10. 10, 195. 95%?

Solution:,

2

11)(

KKXP X 110

200454

1/K2 =0.05 K= ,

21)( KKXP X

95.01

1)10

10( 2 K

KXP

47.420 16.31010

95.0)13.14(

95.0))16.3(47.4(

XP

XP13.14X

13.20987.189

11 - 11.3.4

,Ch19

2

200455

11 - 11.4

200456

11 - 11.4

11.4.1 , that is,

Z p 1-,

p

),(~npq

pNp

p npqZ 2

200457

1-,

1)(2 n

pqZppP

11 -

p)1(

f p()

11.16

200458

spqnp

/ p Z S p 2 / p Z S p 2p p

( ) 1 p/ 2/ 2

11 -

/p Zpqn 2

200459

/ n 2

()

nZp

)2/1(*)2/1( 2/

11 - EX11.8 200712: 40%,

36%. ,144320. 95%,2.5. ?

Solution:n=1443, (11.16),95%

36.0p

200460

,95%

0.0258.

: 95%,33.42%~38.58%: 36%,95%2.58%

0258.036.01443

)5.0)(5.0()96.1(36.02

121

025.0

nZp

3558.03442.0 p

11 - EX11.9 ,

45.24%, 44.90%, 40.78%(359). : . ,95%?

Solution:n=359, ,95%

4078.0p 5922.0q

200461

95%

95%

0508.04078.0359

)5922.0)(4078.0()96.1(4078.0

025.0 n

qpZp

4586.03570.0 p

11 - 11.4.2 (N

11 - 11.5

Concept: 1.

2. , , ,

200463

,,

11 -

d

X Zn

d /2

11.5.1

200464

nZ

d

22 22

()

nZ S

d 2

2 2

2

11 - EX11.10 EX11.5,42.96 ~ 49.04

. 95%2. ?

Solution:

2 22

n

ZX

200465

,(11.19)(S=5.7)

n>=32

, 32,95%2,16

203.312

)7.5()96.1(2

22

2

222

d

SZn

11 - 11.5.2 d,,Z

p, q,

dnpq

Zpp 2

2

22

d

pqZn

200466

p, q, 1. p,q

2. p=1/2, q=1/2p,q

p=1/2, q=1/2, pq,

qp ,

2

22

d

qpZn

2

22 )25.0(

d

Zn

11 - eX11.11 ,

. 95%, 0.03. ?

Solution:

0.03

1.10670009.0

)25.0()96.1()03.0(

)25.0()25.0( 22

2025.0

2

22 Zd

Zn

200467

n>= 1068, 1068

1.10670009.0)03.0( 22

d

n

11 - 11.6

Concept,

EX: ,

200468

, , ( , chi-squaredistribution)

2

11 -

SX X

n

ii

n

2

2

1

1

( )

11.6.1

200469

( ) ( )n S X Xii

n

1

2

2

2

12

)1( n

11 - 11.6.2 (8)

1.0()

0.10

0.12

0.14

0.16f ( )2

v5

200470

0.00

0.02

0.04

0.06

0.08

0.10

0 10 20 30 40 50 60 702

v10

v30

11 - 2.v

v,v,2

3.(v)

vVvE vv 2)(,)(22

200471

(v)

4. , , Z21

)2,(~2 vvNv

2),(~ 2NX

22

XZ

11 - 5.

U ~ , W ~ ,UW,

(U+W) ~

6. n

21v 2

2v

221 vv

n

iX2

200472

6. n

7. n-1

8. X, E(S2)V(S2)

i

iX

1

2

2)1(

Sn

12

)(,)(4

222

nSVSE

11 -

11.5

d.f 0.9952 0.952 0.902 0.102 0.052 0.0052 d.f1 .0000393 .0039321 .0157908 2.70554 3.84146 7.87944 1

2 .0100251 .102587 .210720 4.60517 5.99147 10.5966 2

3 .0717212 .351846 .584375 6.25139 7.81473 12.8381 3

4 .206990 .710721 1.063623 7.77944 9.48773 14.8602 4

5 .411740 1.145476 1.61031 9.23635 11.0705 16.7496 5

200473

5 .411740 1.145476 1.61031 9.23635 11.0705 16.7496 5

6 .675727 1.63539 2.20413 10.6446 12.5916 18.5476 6

17 5.69724 8.67176 10.0852 24.7690 27.5871 35.7185 17

18 6.26481 9.39046 10.8649 25.9894 28.8693 37.1564 18

19 6.84398 10.1170 11.6509 27.2036 30.1435 38.5822 19

20 7.43386 10.8508 12.4426 28.4120 31.4104 39.9968 20

11 - v=5, =0.05 15, (11.18) 1-

210.0

23635.92 10.0,5

1)( 2 222 )2(1P

200474

EX: v =17,=0.10, v=17,

11.19 ,

2)2(1

67176.8295.02

)21.0(1 5871.272

05.02

)21.0(

9.0)5871.2767176.8( 2 P

11 -

f ( )2

11.18

200475

9.2364

=0.10

0 2

v5

11 -

11.19

f ( )2

200476

/2=0.05/2=0.05

27.598.67

v 17

2

11 - 11.6.3

1. S22

2. 2

1

)(1

2

2

n

XXS

n

ii

200477

2. 2

11.20

2S2,()

1))1(

( 2 2,122

2)2(1,1 nn

SnP

1))1()1(

( 2)2(1,1

22

22,1

2

nn

SnSnP

11 - 11.20 )1( 2

f ( )2

200478

1-

2 n1 22 , / n 1 1 22 , /

11 -

( ) ( )n S n S 1 1

2

22

2

2

200479

, / , /n n 1 22

1 1 22

22/1,1

2

22/,1

2 )1()1(

nn

SnSn

11 - EX11.12 16,5cc,

95%?

Solution:

( ) ( )

, / , /

n S n S

n n

1 12

1 22

22

1 1 22

)5(15)5(15 222

200480

95%

, 95%

3.69

11 - 21. ,2S2,

2. n(n), , , ,

2S

21n

1)( 2 1 nE n )1(2)( 2 1 nV n

))1(2,1(~2 nnN

200481

Z2

))1(2,1(~2 1 nnNn

12

11

21 2

22

2

2

nZ

S

nZ

S

11 - 3. ,2

n ,

1-

n

Xn

ii

1

2

2

)(

2

2

2

n

22

2

~

nn

2n ),( 2 2,2

)2(1, nn

200482

1-

2),( 2,)2(1, nn

1)

( 2 2,22

2)2(1, nn

nP

21,

22

2,

2

22

nn

nn

11 - n(n),

Z

)2,(~2

2

nnNn

222

200483

nZ

nZ

21

21 22

11 - EX11.13 ,200,

4, 202, 201, 196, 194. 95%?

Solution:, =200,

(11.30)( )

25.14)(

2

2

n

X

4844.0,1433.11 22

200484

(11.30)( )

: 95%5.12 ~ 117.67

4844.0,1433.11 2 975.0,42

025.0,4

4844.0)25.14(4

1433.11)25.14(4 2

67.11712.5 2

11 - Ex11.14 153.5cm, 150

, 27.85mm.95%

?

Solution:

n=150,

2

))150(2,150(~2150 N

200485

(11.31)

1502

96.11

)85.27(

1502

96.11

)85.27( 2

00.3671.22 2

11 - 11.7

Concept,( )

( )

U

L

200486

11 -

1

1 ZX

200487

XZX

),(X

ZX

11 -

1

XZX ),( XZX

200488

11 -

f x( )

X)1( 11.21

200489

1

x Z X

11 - EX11.15 .

20,000,; , . ,36,21,000, 1,050. ?

Solution:

,S=1050,

050,1S

200490

, 95%(20,712 ~), :20,000,

712,20175645.1000,2136050,1

645.1000,21 n

SZX

11 - EX11.16 .

900,()15,000,6,000. 95%,. 95%?

Solution:

,S=6,000,

200491

,95%(0 ~15,329), : 15,329

Q:?

329,15329645.1000,15900000,6

645.1000,15 n

SZX

11 - eX11.17 ,

360 ,0.7.99%,.99%.

Solution:

,p

3.07.0 qp

200492

99%

,99%(0.644 ~ 1)

024.0360

3.07.0

nqp

p

644.0056.07.0024.033.27.0 pZpp