Sampling Distributionssite.iugaza.edu.ps/ssafi/files/2016/09/توزيعات... · 2016-10-25 ·...
Transcript of Sampling Distributionssite.iugaza.edu.ps/ssafi/files/2016/09/توزيعات... · 2016-10-25 ·...
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
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T1df n
12
250016200S
2400265023502600
191
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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
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122S
224
n 30³m 30³
13
20005001200700
40X
35Y
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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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195
2.4.7
Distribution of the Difference between Two Sample Means from two Independent Non Normal Populations for Large Samples
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5 1 2, , , nx x x 1
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196
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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<
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1 1Sn m
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( ) ( )2 21 22
pn 1 S m 1 S
Sn m 2
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198
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( ) ( )1 2
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+
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n 1 m 1
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+ -
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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
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4 100007510
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13
7.442.88
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0.5 n 48
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75014100
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748.5
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19 40
2037.24.5
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