165

Sampling Distributions

TT

7

167

1.7

1.1.7Sampling

168

2.1.7 Sampling Distribution

3.1.7Paramter and Statistics (Parameter)

(Statistic)

169

2

P

X2Sp1

1

X 2

2S

Pp

4.1.7 n

1X

2X

3X

170

5.1.7Standard Error

n N

nnN

Nn

1153

15N3n315 3375nN

15455

3Nn

3375455

171

2

1 2 3 4 515, 17, 18, 21, 24x x x x x

1 95 195

N

ii

x

N

2

1-

3.16

N

ii

x

N22n

2n = 2

X 2-X

(15,17) 16 9 (15,18) 16.5 6.25 (15,21) 18 1 (15,24)19.5 0.25 (17,18) 17.5 2.25 (17,21) 19 0 (17,24) 20.5 2.25 (18,21) 19.5 0.25 (18,24) 21 4 (21,24) 22.5 12.25 190 37.5

172

X 190 1910XE X

X

37.5 1.9410X

2.7The Sampling Distribution for the Sample Mean

X

X

X

Central Limit Theorem

1.2.7Central Limit Theorem 1 2, , , nX X X

2kkE X

X

X

X

173

3030n

X

X

2.2.7The Mean and Standard Error X

1.

1

1 2, , , nx x x

1X

(1) xE X

2X

N

(2)

2 --1X

N nn N

174

250.05n N

N (3)

X n

350.05n N

- n-1

Nn N

3

1nX

X

X

XX

175

36nX36 6

X33

5

X33X

3X

n X 10 1.581 20 1.118 30 0.913 50 0.707

100 0.500 200 0.354 500 0.224

1000 0.158 3X

30X

X

X

176

4X2

19XE X22 - 3.16 5 - 2

1.94-1 2 5 -1X

N nn N

3.16

1.94X

.

5

2500300

X9n144n

2500E X

X9n 300 100

9X n

X144n 300 25144X n

177

X9n

144n

X166

10005010

25225

50E X

0.05 0.05*1000 50N25 50n

X25n 10 225X n

0.05 0.05*1000 50N225 50n

X225n 22 - 10 1000 - 225

0.587-1 225 1000 -1X

N nn N

178

3.2.7The Sampling Distribution for the Sample Mean from Normal Population

X

x

2X

2

1 2 nx ,x , , x2X

2

n2: ,X N

2

: ,X Nn

:

XZn

: 0,1Z N.

179

7

2500

500100

X

2550

25

2500500

X2500 500 50100X

2550P X

X2500

50X2550P X2550

2550 2500 1

50XZ

n

180

2550 1

1 11 0.8413 0.1587

P X P Z

P Z

15.872550

X

500 50025 20 40025X n n

n

40025X

50100n4

25X 2550 2500 2

25Z

2 1 21 0.9772 0.0228

P Z P Z

181

2.282550

15.872.282550

4.2.7

The Sampling Distribution for the Sample Mean from Not Normal Distribution with Large Sample Size

X

X

XZn

0,1Z N

30n

8

1000200100

950

182

11009701050

X100n2001000, 20100X X

9501000 950 1000950

20 202.5

0.0062

XP X P

P Z

11001000 1100 10001100

20 205

1 5 1 1 0.0000

XP X P

P Z

P Z

Z3.490.99985Z

5 1P Z

1100 1 1 0.0000P X

9701050970 1000 1000 1050 1000970 1050

20 20 201.5 2.5

2.5 1.50.9938 0.0668 0.9270

XP X P

P Z

P Z P Z

183

95010

1.41

30

1 2 30, , ,x x x30

1

1410ii

P x

30n30

30

1 1410 4730 30

ii

xP P X

30n 50

10 30X

50 47 504710 30 10 30

1.640.0505

XP X P

P Z

5.1

184

5.2.7

The Sampling Distribution for the Mean from Normal Population with Unknown Variance

30n

SSnn

XZS n

0,1N

30nS

XS n

TT

185

3.7T T1908W.S. Gosset

TStudent's

T-DistributionTT

T

T=0XZ

nXTS n

ZX

TXS

TZ

ZT

1.3.7Degrees of Freedom T

55n

1

0n

ii

y y

1 2 3 4 5, , , ,d d d d d

William Sealy Gosset1361876 16101937

186

df

–1

2.3.7T T

df

T

T

0E TT.

dfT

2dfVar T

df

df

df

TT

TT

187

3.3.7T

Tdf

Tdf

TT

T

T

;P T df T p df

:

T

;P T T p dfdfT

;T p dfTp

TdfT2;T p dfp0.00050.25

p

; 1 ;T p df T p df

T

T

188

10

TT201.325P TT=1.325

1.72P TT=1.72

2.086P TT=-2.086

2.086 2.845P TT=2.086T=2.845

1.725 1.725P TT=-1.725T=1.725

0.005;20TT0.0050.90;20TT0.90

1.325P TT

205T=1.32

0.101.325 0.10P T10T201.325

T1.72 1 1.72P T P T

1.72P TT

20T=1.721.7250.05

1.72 1 0.05 0.95P T95T201.72

T

2.086 2.086P T P T2.086 0.025P T

189

2.086 2.845P T

2.086 2.845 2.086 2.8450.025 0.0050.020

P T P T P T

1.725 1.725P T

1.725 1.725 1.725 1.725P T P T P T T

1.725 1.725P T P T1.725 1 1.725P T P T

1.725 1.725 1 1.725 1.725

1 2 1.7251 2 0.050.90

P T P T P T

P T

T200.005T=2.845

0.0050.005;20 2.845T

T

0.90;20 0.10;20T TT

200.10T=1.3250.10

0.90;20 0.10;20 1.325T TT= -1.325

0.90

190

11p

; 30 2.457T p;30 1.310T p

T302.4570.01p

; 1 ;T p df T p df

; 30 1 ; 30T p T p 1 ; 30 1.310T p

T1 0.10 0.90p p

3

1 2, , , nx x x230nX

S

XTS n

T1df n

12

250016200S

2400265023502600

191

X16n

2500200 16XTT

15df

24002500 2400 25002400

50 502 2

XP X P

P T P T

T16 1 15df

221.7532.1310.050.025

0.025 2 0.05P T

2P T150.031973

26502500 2650 25002650

50 503

XP X P

P T

T16 1 15df

332.9473.2860.0050.0025

0.0025 3 0.005P T

3P T150.004486

192

235026002350 2500 2500 2600 25002350 2600

50 50 503 2

3 2

XP X P

P T

P T P T

3 1 3

1 31 0.005 0.995

P T P T

P T

2 0.025P T 2350 2600 0.995 0.025 0.970P X

3 2P T150.963541

193

4.7 Distribution of the Difference between Two Sample Means

1.4.7

Distribution of the Difference between Two Sample Means from Independent Normal Populations

44

1 2, , , nx x x 1

21( )2

1 1N1 2, , , my y y

222( )2

2 2N

Y,XX Y-

1 2-2 21 2

n m+ .

( ) ( )1 2

2 21 2

X YZ

n m

- - -=

+

( )Z : N 0,1.

194

21S2

122S

224

n 30³m 30³

13

20005001200700

40X

35Y

700 .

( )P X Y 700- >4:

( ) ( ) ( )

( ) ( )

( )

( ) ( )

( ) ( )

2 2 2 2

X Y 2000 1200 700 2000 1200P X Y 700 P

500 700 500 70040 35 40 35

100P Z P Z 0.7 1 P Z 0.7142.3

1 0.2420 0.7580

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= - =

7000.7580.

195

2.4.7

Distribution of the Difference between Two Sample Means from two Independent Non Normal Populations for Large Samples

5n 30³m 30³

5 1 2, , , nx x x 1

21

1 2, , , my y y222

Y,Xn 30³m 30³

X Y-1 2-2 21 2

n m+

( ) ( )1 2

2 21 2

X YZ

n m

- - -=

+

( )Z : N 0,1»

21S2

122S

225

n 30³m 30³

196

14

23221.52

4050XY

1.25( )P X Y 1.25- >

5

( ) ( ) ( )

( ) ( )

( )

( ) ( )

( ) ( )

2 2 2 2

X Y 23 22 1.25 23 22P X Y 1.25 P

1.5 2 1.5 240 50 40 500.25P Z P Z 0.68 1 P Z 0.68

0.3691 0.7517 0.2483

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= - =

1.250.2483

197

3.4.7

Distribution of the Difference between Two Sample Means from two Independent Normal Populations for Small Samples

6n 30<m 30<

6

1 2, , , nx x x1

21( )2

1 1N1 2, , , my y y

222( )2

2 2N21

22

2 21 2=

Y,XX Y-T

1 2-2pS

(Pooled Variance)

( ) ( )1 2

p

X YT

1 1Sn m

- - -=

+

Tdf n m 2= + -2pS

( ) ( )2 21 22

pn 1 S m 1 S

Sn m 2

- + -=

+ -

198

21

22

2 21 2¹

( ) ( )1 2

2 21 2

X YT

S Sn m

- - -=

+

T:

( ) ( ) ( )2 21 2

2 21 2

12 2S S

2 n mS Sn mdf

n 1 m 1

-é ùê úê ú= + +ê ú

- -ê úê úë û

( )1 1- -n , m15

657214

201825Y,X

4

199

( )P X Y 5- <

( ) ( ) ( )( )2 22p

p

14 1 20 18 1 25S 527.5

14 18 2S 527.5 22.97

- ´ + -= =

+ -

= @

df 14 18 2 30= + - =6

( ) ( ) ( ) ( )

( )( )df 30

X Y 65 72 4 65 72P X Y 4 P

1 1 1 122.97 22.9714 18 14 18

P T 1.34

1 P T 1.341 0.10 0.90

=

æ ö÷ç ÷ç ÷- - -ç - - ÷ç ÷- < = <ç ÷ç ÷ç ÷÷ç + + ÷ç ÷çè ø

= <

= - >

@ - =

40.901.34P T300.904846

200

5.71 X495036

4812 2

1400480

X

1040160 X

X 12

3

752022

78 70 7280

4 100007510

100

1000

201

5

6520

60 8 11

6 70

3506

5.5 5

7

T 0.975 , 20 T 0.01,18 T 0.05 ,10 T 0.995 , 30

8 p

T p , 20 1.725 T p ,16 2.12

T p ,15 1.341 T p , 25 2.787

202

9 T20P T 2.528

P T 1.725

P 2.086 T 2.528

10

X

n 50 40, 6

n 12 40, 6

n 5040, 6

n 1240, 6

11 10X

1.5

X

12 Xn 100

203

X

13

7.442.88

X

0.5 n 48

n 4

n 4

14 X

n

X

X

15

100080

204

641010

16

75014100

748.5 n 100

X

748.5

17 XT

S n

T

n 9

n 9

n 44

n 44

205

18 XTS n

XZn

10

n 15

X 48.2S 8.2

10n 40X 48.2

n 40 X 48.2S 8.2

19 40

2037.24.5

2037.2

206

20 100.01

16

X

16

10.005

0.002 X0.002

21

6710585

22

309

10 35 40

207

3238 –36

23 A6.5

0.9B

60.836A

49B

24 A150

B17080A

3050B20

40