ce's for ch7 - 10
TRANSCRIPT
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Example 7-2 Central Limit Theorem
n 40
mu 5
sigma^2 =(6-4)^2/12 = 1/3 1/3
mubar 5
sigma^2/n =1/(3*B13) = 1/120 1/120
Suppose that a random variable X has a continuous unif
Find the distribution of the sample mean of a random sa
1 2, 4 x 6
0, otherwise f x
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rm distribution:
mple of size n = 40.
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given
n mean () Standard Deveation ( n mean () Standard Deveation ()
16 5000 40 25 5050 30
Find
P(x2bar-x1bar)>25
z = x2bar-x2bar-(u1-u2)/(sample variance)
z -2.14373
= 1-0.0161 0.983823
Old New
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GIVEN DATA
12.8 mean (mu) Xbar sigma
9.4 11.04 11.04 1.967627
8.7
11.6 mu = xbar
13.19.8
14.1
8.5
12.1
10.3
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given
n Xbar stdev sigma(xbar)^2
data 41.6 10 41.924 0.284105 0.089841836
41.48 standard error =~ 2%
42.34
41.95
41.86
42.18
41.72
42.26
41.81
42.04
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x1 11.96 Xbar mda = 1/Xbar
x2 5.03 21.64625 0.046197
x3 67.4
x4 16.07
x5 31.5
x6 7.73
x7 11.1
x8 22.38
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Example8-4Concentration
1.23 0.04
1.33 0.04
0.04 0.044
0.044 0.05
1.2 0.1
0.27 0.150.49 0.16
0.19 0.17
0.83 0.18
0.81 0.19
0.71 0.19
0.5 0.19
0.49 0.21
1.16 0.25
0.05 0.27
0.15 0.27
0.19 0.27
0.77 0.28
1.08 0.34
0.98 0.34
0.63 0.34
0.56 0.34
0.41 0.4
0.73 0.41
0.59 0.43
0.34 0.49
0.34 0.490.84 0.49
0.5 0.5
0.34 0.5
0.28 0.52
0.34 0.56
0.75 0.56
0.87 0.59
0.56 0.63
0.17 0.65
0.18 0.71
0.19 0.730.04 0.75
0.49 0.77
1.1 0.81
0.16 0.83
0.1 0.84
0.21 0.86
0.86 0.87
0.52 0.94
0
0.2
0.4
0.6
0.8
1
1.2
1.4
0 10 20 30
y = 0.0
R
-0.2
0
0.2
0.4
0.6
0.8
1
1.2
1.4
0 10 20 30
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0.65 0.98
0.27 1.08
0.94 1.1
0.4 1.16
0.43 1.2
0.25 1.23
0.27 1.33
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40 50 60
Series1
221x - 0.0716
= 0.958
40 50 60
Series1
Linear (Series1)
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19.8 7.5 Example8-5Load
15.4 7.9
11.4 8.8
19.5 10.1
10.1 11.4
18.5 11.4
14.1 11.48.8 11.9
14.9 11.9
7.9 12.7
17.6 13.6
13.6 14.1
7.5 14.9
12.7 15.4
16.7 15.4
11.9 15.4
15.4 15.8
11.9 16.7
15.8 17.6
11.4 18.5
15.4 19.5
11.4 19.8
0
5
10
15
20
25
0 5 10
y = 0.5412x + 7.4896
R = 0.9781
0
5
10
15
20
25
0 5 10 15 20 2
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15 20 25
Series1
5
Series1
Linear (Series1)
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A Ok
B Not Ok your Ho needs to be an equality.
C Not Ok you can't use the xbar for your hypothesis
D Not Ok you can't have two equality in your hypothesis
E Not Ok You can't use the sample to test you hypothesis
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Given
dist = normal
mu 100
STD 2
n = 9
Test
H0 = 100
H1 != 100
A
acceptance reagion
98.5
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Ex9-57
421
452.6 Column1 1) mean 300
456.1 2) H0 =300
494.6 Mean 325.4963 3) H1 !=300
373.8 Standard E 38.25629 4) T statistic T0=xbar-/(S/sqrt(n))447.8 Median 290.9 Hypothesis on the mean with variance unkno
687.6 Mode 296
705.7 Standard D 198.7855 n 27
879 Sample Va 39515.69 xbar 325.4963
88.8 Kurtosis 1.304034 S 198.7855
90.5 Skewness 1.097014 300
110.7 Range 797.3
96.4 Minimum 81.7 5) Rejection Criteria
81.7 Maximum 879 T0>t(alpha/2,n-1)
102.4 Sum 8788.4 6) T0 0.66646
296 Count 27 0.025
273 t 2.056 book
268 2.055529 excel
227.5 T0 T
279.3 0.66646 < 2.056
241
296 7) Fail to reject the null hypothesis that =300
317 at the significacne level =0.05
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290.9
256.5
258.5 0.5 < P-value < 0.8
296
P-value
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n.
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example 9-12
Number of defects Observed Frequency
0 32
1 152 9
3 4
given
H0: Poisson distrobution
n 60
lamda 0.75 mean defect frequency Number of defects
Using Pag 704 equ for poisson dist 0
1
X Poisson = P(Xi) 2
0 0.472367 3
1 0.354275
2 0.132853
3 0.040505
expected freq in the last cell is less then 3 so combine last 2 cells
degrees of freedom k-p-1 3 - 1 -1 1
7 step Hypo Procedure
1) parameter of interest? distrubution of the defects s what was assumed i.e. Poss
2) H0: distribution is poisson
3) alternate Hypo: not Poisson
4) test Statistic equ = 9-47
5) H0 if Pvalue less then significance level 95%
6) computations chi0^2 = 2.96278
7) chi^2(,1) 2.71 3.84
P
0.1
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Conclusions
the information follows Poisson
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Observed Frequency expected frequency
Ei
32 28.34199316 32
15 21.25649487 15
9 7.971185578 9
4 2.430326385
n
32 28.34199
15 21.25649
9 10.4015
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modofied table for expected frequency
28.34199
21.25649
7.971186
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cat1 cat2 Sp^2 =((n1-1)s1^2 + (n2-1)s2^2))/(n1-n2-2)
1 91.5 89.19 Sp^3 7.294654
2 94.18 90.95 Sp 2.7008623 92.18 90.46 t0 = =(xbar1-xbar2)/(2.70*sqrt(1/n1+1/n2))
4 95.39 93.21 t0 = -0.35359
5 91.79 97.19
6 89.07 97.04 Conclusion we do not have strong evidance to conclude t
7 94.72 91.07
8 89.21 92.75
xbar 92.255 92.7325
s 2.385019 2.983453
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hat catalyst 2 results in a mean yield that differs fro the mfean yeald whedn catalyst 1 is used.
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EXAMPLE 10-5
cat1 cat2 Column1
x1bar x2bar 1 91.5 89.19
92.255 92.733 2 94.18 90.95 Mean 92.255
3 92.18 90.46 Standard E 0.843231
s1 s2 4 95.39 93.21 Median 91.985
2.39 2.98 5 91.79 97.19 Mode #N/A6 89.07 97.04 Standard D 2.385019
n1 n2 7 94.72 91.07 Sample Va 5.688314
8 8 8 89.21 92.75 Kurtosis -1.33448
Skewness -0.11444
given Variances of the population equal Range 6.32
Minimum 89.07
Maximum 95.39
1 difference on mean mu (mu1 and mu2) Sum 738.04
Count 8
2 H0: mu1 = mu2
3 H1: mu1 != mu2
4 test statistics
t0 = eq10.14 in the text book 0.728914
5 rejection criterion to>t(a;pha/2,dof)
6 compution
sp^2 - pooled bariance 7.29625
sp - pooled variance 2.701157
T0 = -0.35392to>t(a;pha/2,dof) 2.145
failed to reject
7 Concolution
mu1 = mu2
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Column2
Mean 92.7325
Standard E 1.05481
Median 91.91
Mode #N/AStandard D 2.983453
Sample Va 8.900993
Kurtosis -0.82782
Skewness 0.732691
Range 8
Minimum 89.19
Maximum 97.19
Sum 741.86
Count 8
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Example10-6Mmple10-6Rural
1 3 48 alpha 0.05
2 7 44 alpha / 2 0.025
3 25 40 v num 863.7068
4 10 38 v denm 65.45433
5 15 33 v 13.19556 ~ 13
6 6 21 t(alpha/2,13) 2.16 from table7 12 20
8 25 12 t0 =(xbar1-xbar2)/(sqrt(S1^2/n1+s2
9 15 1 t0 -2.76694
10 7 18
xbar 12.5 27.5
s 7.633988 15.34963 Clonclusion
t(alpha) > t0 reject H0
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2/n2))
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EXAMPLE 10-6
Example1 Example10-6Rural
3 48 Column1 Column2
7 44
25 40 Mean 12.5 Mean 27.510 38 Standard E 2.414079 Standard E 4.853979
15 33 Median 11 Median 27
6 21 Mode 7 Mode #N/A
12 20 Standard D 7.633988 Standard D 15.34963
25 12 Sample Va 58.27778 Sample Va 235.6111
15 1 Kurtosis -0.4382 Kurtosis -0.98883
7 18 Skewness 0.76704 Skewness -0.28814
Range 22 Range 47
Minimum 3 Minimum 1
Maximum 25 Maximum 48
Sum 125 Sum 275
Count 10 Count 10
0.015827
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EXAMPLE 10-10
Girder Example10-10Karlsruhe Example10-10Lehigh difference dbar
S1/1 1.186 1.061 0.125 0.273889
S2/1 1.151 0.992 0.159
S3/1 1.322 1.063 0.259 stdS4/1 1.339 1.062 0.277 0.135099
S5/1 1.2 1.065 0.135
S2/1 1.402 1.178 0.224 t0
S2/2 1.365 1.037 0.328 6.081939
S2/3 1.537 1.086 0.451
S2/4 1.559 1.052 0.507 p value
0.000295
n= 9
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Given
n 15
xbar 0.27
s 0.0041
CI 0.95
alpha 0.05 stdev upper bound
EQ 8-19 CI on the variance
Sigma^2 = (n-1)s^2/((chi^2)(1-alpha,n-1))
alpha 0.05
n-1 14s^2 1.68E-05
one sided CI for the variance for Upper bound Sigma^2 3.58E-05 6.570631
sigma 0.005985
A set of 15 samples are taken randomly from a pr
The data is: Xbar = 0.27, s = 0.0041
Calculate a 95% Confidence Interval on the stdev
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oduction line and measured.
upper bound?
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0.58137 Column1
0.5872
0.57997 Mean 0.606727
0.57982 Standard Error 0.028023
0.57935 Median 0.578755
0.56583 Mode 0.579350.55753 Standard Deviation 0.33628
0.57867 Sample Variance 0.113084
0.57618 Kurtosis 74.38434
0.57884 Skewness 8.213917
0.59917 Range 3.893755
0.59541 Minimum 0.010745
0.57679 Maximum 3.9045
0.57534 Sum 87.36868
0.58261 Count 144
0.57541
0.56571
0.5783
0.58827
0.56514
0.57008
0.58766
0.56788
0.58095
0.58036
0.58904
0.579350.57223
0.58318
0.57899
0.55758
0.57511
0.56764
0.59709
0.56251
0.58382
0.57578
0.60585
0.5867
3.9045
0.58456
0.57591
0.58856
0.57614
0.58574
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
0 20 40 60 80
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0.59158
0.59905
0.56593
0.58821
0.56655
0.59519
0.561040.58841
0.58538
0.58541
0.5878
0.595
0.57985
0.58157
0.60112
0.58941
0.59512
0.58867
0.5635
0.59146
0.57064
0.35369
0.57508
0.58803
0.59457
0.59218
0.57766
0.578470.57615
0.56708
0.58341
0.58712
0.57278
0.60098
0.56946
0.32789
0.5938
0.59544
0.585710.58059
0.5708
0.59438
0.6007
0.59804
0.58286
0.56949
0.52125
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0.59138
0.57085
0.57387
0.574
0.14238
0.58808
1.04280.56814
0.59273
0.58896
0.29362
0.53753
0.58077
0.60482
0.57713
0.58072
0.57208
0.56483
0.57193
0.57675
0.56983
0.58618
0.57666
0.57151
0.56379
0.56989
0.57923
0.582360.56965
0.5705
0.57931
0.5433
0.56562
0.56632
0.55786
0.57895
0.57005
2.6464
0.57750.57469
0.58252
0.55127
0.57797
0.58799
0.57702
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0.52125
0.52896
0.53753
0.5433
0.55127
0.556460.55753
0.55758
0.55786
0.56104
0.56251
0.5635
0.56379
0.56483
0.56514
0.56562
0.56571
0.56583
0.56593
0.56632
0.56655
0.56708
0.56764
0.56788
0.56814
0.56946
0.569490.56965
0.56983
0.56989
0.57005
0.57008
0.5705
0.57064
0.5708
0.57085
0.57151
0.57193
0.57208
0.57223
0.57278
0.57387
0.574
0.57469
0.57477
100 120 140 160
Series1
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0.57508
0.57511
0.57534
0.57541
0.57578
0.57591
0.576140.57615
0.57618
0.57638
0.57666
0.57675
0.57679
0.57702
0.57713
0.57735
0.5775
0.57766
0.57797
0.5783
0.57847
0.57867
0.57884
0.57895
0.57899
0.57923
0.57931
0.579350.57935
0.57982
0.57985
0.57997
0.58036
0.58059
0.58072
0.58077
0.58095
0.58137
0.581570.58165
0.58236
0.58252
0.58261
0.58286
0.58318
0.58341
0.58382
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0.58456
0.58538
0.58541
0.58571
0.58574
0.58618
0.58670.58712
0.5872
0.58766
0.5878
0.58799
0.58803
0.58808
0.58821
0.58827
0.58841
0.58856
0.58867
0.58896
0.58904
0.58941
0.59138
0.59146
0.59158
0.59218
0.59273
0.59380.59438
0.59457
0.595
0.59512
0.59519
0.59541
0.59544
0.59709
0.59804
0.59905
0.599170.6007
0.60098
0.60112
0.60482
0.60585
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Column1
Mean 0.578284
Standard Error 0.001159
Median 0.578895
Mode 0.57935Standard Deviation 0.013513
Sample Variance 0.000183
Kurtosis 2.990452
Skewness -1.04009
Range 0.0846
Minimum 0.52125
Maximum 0.60585
Sum 78.64665
Count 136
0.4 Bin Frequency
0.45 0.4 0
0.5 0.45 0
0.55 0.5 0
0.6 0.55 4
0.65 0.6 127
0.7 0.65 5
0.75 0.7 0
0.75 0
More 0
0.51
0.52
0.53
0.54
0.55
0.56
0.57
0.58
0.59
0.6
0.61
0.62
0 20 40 60 80 100 120 140
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0
20
40
60
80
100
120
140
0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75
Histogram
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160
Series1
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More
Frequency
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Vt0 data from one wafer
1. choose some smaller wafer regions and calculate u, s, and Z for each.
2. copy the whole thing onto a 2nd sheet, and on that sheet eliminate outliers, then compare results.
0.58856 0.59500 0.59218 0.58059
0.58827 0.55758 0.57614 0.57985 0.57766 0.57080
0.57618 0.56514 0.57511 0.58574 0.58157 0.57847 0.59438 0.568140.58137 0.57884 0.57008 0.56764 0.59158 0.60112 0.57615 0.60070 0.59273
0.58720 0.59917 0.58766 0.59709 0.59905 0.58941 0.56708 0.59804 0.58896
0.57997 0.59541 0.56788 0.56251 0.56593 0.59512 0.58341 0.58286
0.57982 0.57679 0.58095 0.58382 0.58821 0.58867 0.58712 0.56949 0.53753
0.57935 0.57534 0.58036 0.57578 0.56655 0.56350 0.57278 0.52125 0.58077
0.56583 0.58261 0.58904 0.60585 0.59519 0.59146 0.60098 0.59138 0.60482
0.55753 0.57541 0.57935 0.58670 0.56104 0.57064 0.56946 0.57085 0.57713
0.57867 0.56571 0.57223 0.58841 0.35369 0.57387 0.58072
0.57830 0.58318 0.58456 0.58538 0.57508 0.59380 0.57400 0.57208
0.57899 0.57591 0.58541 0.58803 0.59544 0.56483
0.58780 0.59457 0.58571 0.58808
Population small n larger n even larger
Count 137 count 6.00000 60.00000 120.00000
u 0.576 Xbar 0.57548 0.57733 0.57621
S 0.023685 s 0.01365 0.03189 0.02466
Z 0 z -0.08226 0.34679 -0.02678
s 0.58856 0.59500 0.59218 0.58059
0.58827 0.55758 0.57614 0.57985 0.57766 0.57080 1.04280
0.57618 0.56514 0.57511 0.58574 0.58157 0.57847 0.59438 0.56814
0.58137 0.57884 0.57008 0.56764 0.59158 0.60112 0.57615 0.60070 0.59273
0.58720 0.59917 0.58766 0.59709 0.59905 0.58941 0.56708 0.59804 0.58896
0.57997 0.59541 0.56788 0.56251 0.56593 0.59512 0.58341 0.58286 0.29362
0.57982 0.57679 0.58095 0.58382 0.58821 0.58867 0.58712 0.56949 0.53753
0.57935 0.57534 0.58036 0.57578 0.56655 0.56350 0.57278 0.52125 0.58077
0.56583 0.58261 0.58904 0.60585 0.59519 0.59146 0.60098 0.59138 0.604820.55753 0.57541 0.57935 0.58670 0.56104 0.57064 0.56946 0.57085 0.57713
0.57867 0.56571 0.57223 0.58841 0.35369 0.32789 0.57387 0.58072
0.57830 0.58318 0.58456 0.58538 0.57508 0.59380 0.57400 0.57208
0.57899 0.57591 0.58541 0.58803 0.59544 0.14238 0.56483
0.58780 0.59457 0.58571 0.58808
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0.58856 0.59500 0.59218 0.58059
0.58827 0.55758 0.57614 0.57985 0.57766 0.57080 1.04280
0.57618 0.56514 0.57511 0.58574 0.58157 0.57847 0.59438 0.56814
0.58137 0.57884 0.57008 0.56764 0.59158 0.60112 0.57615 0.60070 0.592730.58720 0.59917 0.58766 0.59709 0.59905 0.58941 0.56708 0.59804 0.58896
0.57997 0.59541 0.56788 0.56251 0.56593 0.59512 0.58341 0.58286 0.29362
0.57982 0.57679 0.58095 0.58382 0.58821 0.58867 0.58712 0.56949 0.53753
0.57935 0.57534 0.58036 0.57578 0.56655 0.56350 0.57278 0.52125 0.58077
0.56583 0.58261 0.58904 0.60585 0.59519 0.59146 0.60098 0.59138 0.60482
0.55753 0.57541 0.57935 0.58670 0.56104 0.57064 0.56946 0.57085 0.57713
0.57867 0.56571 0.57223 3.90450 0.58841 0.35369 0.32789 0.57387 0.58072
0.57830 0.58318 0.58456 0.58538 0.57508 0.59380 0.57400 0.57208
0.57899 0.57591 0.58541 0.58803 0.59544 0.14238 0.56483
0.58780 0.59457 0.58571 0.58808
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49/52
0.57193 Xbar small n
0.57675 0.57931 sigma / sqr 60.56983 0.54330 0.58252 Mean mu 0.57664 0.57548
0.58618 0.56562 0.55127 Standard Dev sigma 0.02344 0.013652
0.57666 0.56632 0.57797 0.57477 Median 0.57884 0.56898
0.57151 0.55786 0.58799 0.57638
0.56379 0.57895 0.57702 0.52896
0.56989 0.57005 0.58165 0.57735 Z 0.00000 -0.12207
0.57923
0.58236 0.57750 0.55646 z(alpha/2*sqrt(n)
0.56965 0.57469 lower bound
0.57050 upper bound
n
0.57193
0.57675 0.57931
0.56983 0.54330 0.58252
0.58618 0.56562 0.55127
0.57666 0.56632 0.57797 0.57477
0.57151 0.55786 0.58799 0.57638
0.56379 0.57895 0.57702 0.52896
0.56989 0.57005 0.58165 0.577350.57923 2.64640 0.01075
0.58236 0.57750 0.55646
0.56965 0.57469
0.57050
Population Information
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8/2/2019 CE's for Ch7 - 10
50/52
0.57193 Xbar small n
0.57675 0.57931 sigma / sqr 6
0.56983 0.54330 0.58252 Mean mu 0.60673 0.575480.58618 0.56562 0.55127 Standard Dev sigma 0.33628 0.013652
0.57666 0.56632 0.57797 0.57477 Median 0.57876 0.56898
0.57151 0.55786 0.58799 0.57638
0.56379 0.57895 0.57702 0.52896
0.56989 0.57005 0.58165 0.57735 Z 0.00000 -0.22763
0.57923 2.64640 0.01075
0.58236 0.57750 0.55646
0.56965 0.57469
0.57050
Population Information
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8/2/2019 CE's for Ch7 - 10
51/52
alpha
medium n large n even larger n
20 60 1200.58074 0.57733 0.57621
0.011988005 0.03189 0.02466
0.57990 0.58066 0.57911
0.781120538 0.227219188 -0.201273464
effect of sample size
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52/52
medium n large n even larger n
20 60 120
0.58074 0.62863 0.610080.011988005 0.43242 0.36312
0.57990 0.58066 0.57897
-0.345609353
effect of sample size