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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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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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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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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