Estimation Chapter 7

EstimationChapter 7Farewell! Thou are too dear for my possessingAnd like enough thou knowst thy estimate.William Shakespeare, Sonnet 87

Topics and Goals for Chapter 7Unbiased point estimators for the population mean:sample meansample mediansample trimmed meanInterval estimate of a population meanConfidence interval for a proportionSample size and confidence intervalsWhat to do when the population variance is unknownConfidence intervals with the Students t-distributionAssumptions for the Students t-distribution

Unbiased Estimators of the Population MeanAn estimator of the mean, hat ( with a caret over it) is unbiased if E(uhat) = , that is if the long run average of hat equals the population mean.The sample mean, xbar, is an unbiased estimator of ;The sample median, xm, is an unbiased estimator of ;The trimmed sample mean (doesnt take top 10%, bottom 10% of values) is an unbiased estimator of .Any linear combination of the sample values, divided by the number of values, is an unbiased estimator of (see Problem 7.8, where the middle of the range is used to estimate ).

Efficient Estimators of the Population MeanThe most efficient estimator of the population mean is that which will give an estimate with the smallest standard deviation.Example: Problems 7.1, 7.2, 7.8 (Electronic Reserve).

Confidence Interval for Population Mean-ISuppose we measure a sample mean; how close is this value to the population mean?If we know the standard deviation for the population, this is a straightforward problem; if we dont, its more complicated--for now well suppose we know the value of , the population standard deviationThe population distribution for the sample means, sample size N, will approach a normal distribution, mean , and standard deviation of the mean, mean = /N, as N gets large (practically, for N greater than about 10 to 20).

Confidence Interval for Population Mean-IIThere is a 95% probability that the measured value of the sample mean, xbar, lies within the range -1.96 mean to + 1.96 mean (See Board demo)This corresponds to the inequality -1.96 mean xbar +1.96 mean With a little manipulation the inequality above can be changed to the one for the confidence interval (CI) xbar-1.96 mean xbar +1.96 mean where mean = /N. Interpretation: 95% of the trials (in the long run) will give values of xbar within limits.(Concepts example)

Confidence Interval for Population Mean-IIIGeneral Case: Confidence Interval (CI) for level (1-)*100 % (e.g. = 0.05 corresponds to 95% level)Then the CI (1-) is given by xbar - z (1-/2) mean xbar + z (1-/2) mean , where mean = / N and z (1-/2) is the z-score for the (1-/2) centile (see board diagram):

Confidence Level

z (1-(/2)

90 %

1.645

95%

1.960

99%

2.575

Interpretation of Confidence IntervalThe diagram to the left is from the Concepts StatPlus add-in. , the true mean salary, equals $5600;The 95% CI for given and N runs from

Chart1

5101.45865550171

5655.819364371

5174.29245617592

5728.65316504412

5382.35071855683

5936.7114274253

5357.34114033544

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5234.13704522895

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5331.41995387186

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5007.9710597635007.971059763

5562.33176863135562.3317686313

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5572.44145580465572.4414558046

5336.65711910699

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5548.664397651211

6103.025106519411

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Confidence Intervals for Sample Means

Title Page

Statistical Concepts

for Excel 97

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Box Plots

Probability Density Function

Probability Simulation

Central LimitTheorem

ConfidenceIntervals

HypothesisTesting

t Distribution

Chi-sq Distribution

F Distribution

TESTS AND SIMULATIONS

DISTRIBUTIONS

TIME SERIES

1 Parameter Exponential Smoothing


Box Plot

DataBox PlotsXOuter Y1Inner Y1Stem X1XBBox1XFQ0

00.0000.0000.0000.00.0001.50

Box Plot Parameters0.00.000

VariableData0.00.000XTQ0

Outer Upper Fence0.0000.00.0001.50

Inner Upper Fence0.0000.00.000

Upper Whisker0.0000.00.000XM0

Maximum0.0000.00.0001.50

Third Quartile0.0000.00.000

Median0.0000.00.000XLW0

First Quartile0.0000.00.0001.00

Minimum0.0000.00.000

Lower Whisker0.0000.00.000XUW0

Inner Lower Fence0.0000.00.0001.00

Outer Lower Fence0.0000.00.000

Interquartile Range0.000XILF0

0.50

1.50

XIUF0

0.50

1.50

XOLF0

0.50

1.50

XOUF0

0.50

1.50

Outliers

2.50

2.50

0

2.50

0

2.40

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Box Plot

000000000000000

00000000

0

0

0

0

0

0

0

0

0

0

0

0

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Box1

Inner Y1

Outer Y1

Outliers

Outliers

Density Function


-1.86

1.76

114.00

476.00

Probability that the value is between -1.86 & 1.76 :92.9%

Reset

Box Plot

Help

Return to Title Page

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Density Function

0.00443184840

0.00514264090

0.00595253240

0.00687276670

0.00791545160

0.00909356250

0.01042093480

0.01191224360

0.01358296920

0.01544934710

0.01752830050

0.01983735440

0.02239453030

0.02521821990

0.02832703770

0.03173965180

0.03547459280

0.03955004160

0.0439835960

0.04879201860

0.05399096650

0.05959470610

0.06561581480

0.07206487430.0720648743

0.07895015830.0789501583

0.08627731880.0862773188

0.09404907740.0940490774

0.10226492460.1022649246

0.11092083470.1109208347

0.12000900070.1200090007

0.12951759570.1295175957

0.13943056640.1394305664

0.14972746560.1497274656

0.16038332730.1603833273

0.1713685920.171368592

0.18264908540.1826490854

0.1941860550.194186055

0.20593626870.2059362687

0.2178521770.217852177

0.22988214070.2298821407

0.24197072450.2419707245

0.25405905650.2540590565

0.26608524990.2660852499

0.27798488610.2779848861

0.28969155280.2896915528

0.30113743220.3011374322

0.31225393340.3122539334

0.32297235970.3229723597

0.33322460290.3332246029

0.3429438550.342943855

0.35206532680.3520653268

0.36052696250.3605269625

0.36827014030.3682701403

0.37524034690.3752403469

0.38138781550.3813878155

0.38666811680.3866681168

0.3910426940.391042694

0.39447933090.3944793309

0.39695254750.3969525475

0.39844391410.3984439141

0.39894228040.3989422804

0.39844391410.3984439141

0.39695254750.3969525475

0.39447933090.3944793309

0.3910426940.391042694

0.38666811680.3866681168

0.38138781550.3813878155

0.37524034690.3752403469

0.36827014030.3682701403

0.36052696250.3605269625

0.35206532680.3520653268

0.3429438550.342943855

0.33322460290.3332246029

0.32297235970.3229723597

0.31225393340.3122539334

0.30113743220.3011374322

0.28969155280.2896915528

0.27798488610.2779848861

0.26608524990.2660852499

0.25405905650.2540590565

0.24197072450.2419707245

0.22988214070.2298821407

0.2178521770.217852177

0.20593626870.2059362687

0.1941860550.194186055

0.18264908540.1826490854

0.1713685920.171368592

0.16038332730.1603833273

0.14972746560.1497274656

0.13943056640.1394305664

0.12951759570.1295175957

0.12000900070.1200090007

0.11092083470.1109208347

0.10226492460.1022649246

0.09404907740.0940490774

0.08627731880.0862773188

0.07895015830

0.07206487430

0.06561581480

0.05959470610

0.05399096650

0.04879201860

0.0439835960

0.03955004160

0.03547459280

0.03173965180

0.02832703770

0.02521821990

0.02239453030

0.01983735440

0.01752830050

0.01544934710

0.01358296920

0.01191224360

0.01042093480

0.00909356250

0.00791545160

0.00687276670

0.00595253240

0.00514264090

0.00443184840

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Distribution Data

-30.00443184840-1.86

-2.950.005142640901.76

-2.90.00595253240

-2.850.00687276670

-2.80.00791545160

-2.750.00909356250

-2.70.01042093480

-2.650.01191224360

-2.60.01358296920

-2.550.01544934710

-2.500.01752830050

-2.450.01983735440

-2.400.02239453030

-2.350.02521821990

-2.300.02832703770

-2.250.03173965180

-2.200.03547459280

-2.150.03955004160

-2.100.0439835960

-2.050.04879201860

-2.000.05399096650

-1.950.05959470610

-1.900.06561581480

-1.850.07206487430.0720648743

-1.800.07895015830.0789501583

-1.750.08627731880.0862773188

-1.700.09404907740.0940490774

-1.650.10226492460.1022649246

-1.600.11092083470.1109208347

-1.550.12000900070.1200090007

-1.500.12951759570.1295175957

-1.450.13943056640.1394305664

-1.400.14972746560.1497274656

-1.350.16038332730.1603833273

-1.300.1713685920.171368592

-1.250.18264908540.1826490854

-1.200.1941860550.194186055

-1.150.20593626870.2059362687

-1.100.2178521770.217852177

-1.050.22988214070.2298821407

-1.000.24197072450.2419707245

-0.950.25405905650.2540590565

-0.900.26608524990.2660852499

-0.850.27798488610.2779848861

-0.800.28969155280.2896915528

-0.750.30113743220.3011374322

-0.700.31225393340.3122539334

-0.650.32297235970.3229723597

-0.600.33322460290.3332246029

-0.550.3429438550.342943855

-0.500.35206532680.3520653268

-0.450.36052696250.3605269625

-0.400.36827014030.3682701403

-0.350.37524034690.3752403469

-0.300.38138781550.3813878155

-0.250.38666811680.3866681168

-0.200.3910426940.391042694

-0.150.39447933090.3944793309

-0.100.39695254750.3969525475

-0.050.39844391410.3984439141

-0.000.39894228040.3989422804

0.050.39844391410.3984439141

0.100.39695254750.3969525475

0.150.39447933090.3944793309

0.200.3910426940.391042694

0.250.38666811680.3866681168

0.300.38138781550.3813878155

0.350.37524034690.3752403469

0.400.36827014030.3682701403

0.450.36052696250.3605269625

0.500.35206532680.3520653268

0.550.3429438550.342943855

0.600.33322460290.3332246029

0.650.32297235970.3229723597

0.700.31225393340.3122539334

0.750.30113743220.3011374322

0.800.28969155280.2896915528

0.850.27798488610.2779848861

0.900.26608524990.2660852499

0.950.25405905650.2540590565

1.000.24197072450.2419707245

1.050.22988214070.2298821407

1.100.2178521770.217852177

1.150.20593626870.2059362687

1.200.1941860550.194186055

1.250.18264908540.1826490854

1.300.1713685920.171368592

1.350.16038332730.1603833273

1.400.14972746560.1497274656

1.450.13943056640.1394305664

1.500.12951759570.1295175957

1.550.12000900070.1200090007

1.600.11092083470.1109208347

1.650.10226492460.1022649246

1.700.09404907740.0940490774

1.750.08627731880.0862773188

1.800.07895015830

1.850.07206487430

1.900.06561581480

1.950.05959470610

2.000.05399096650

2.050.04879201860

2.100.0439835960

2.150.03955004160

2.200.03547459280

2.250.03173965180

2.300.02832703770

2.350.02521821990

2.400.02239453030

2.450.01983735440

2.500.01752830050

2.550.01544934710

2.600.01358296920

2.650.01191224360

2.700.01042093480

2.750.00909356250

2.800.00791545160

2.850.00687276670

2.900.00595253240

2.950.00514264090

3.000.00443184840


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Target


Vertical Deviation 50 (1-tailed)

mu 50 (2-tailed)


Help

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t-Distribution Data


Degrees of Freedom:15

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t-Distribution Data

0.00117899970.0001338302

0.00197580220.0003525957

0.00330731210.0008726827

0.00551544590.0020290481

0.00913518390.0044318484

0.01497302250.0090935625

0.02418442440.0175283005

0.0383112950.0317396518

0.05920773160.0539909665

0.08875326720.0862773188

0.12826094490.1295175957

0.17758247320.1826490854

0.23412477290.2419707245

0.2922619750.3011374322

0.34375457030.3520653268

0.37951657760.3866681168

0.39235316280.3989422804

0.37951657760.3866681168

0.34375457030.3520653268

0.2922619750.3011374322

0.23412477290.2419707245

0.17758247320.1826490854

0.12826094490.1295175957

0.08875326720.0862773188

0.05920773160.0539909665

0.0383112950.0317396518

0.02418442440.0175283005

0.01497302250.0090935625

0.00913518390.0044318484

0.00551544590.0020290481

0.00330731210.0008726827

0.00197580220.0003525957

0.00117899970.0001338302

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Standard Normal

t Distribution

Chi-sq DF

-4.000.00117899970.000133830215

-3.750.00197580220.0003525957

-3.500.00330731210.0008726827

-3.250.00551544590.0020290481

-3.000.00913518390.0044318484

-2.750.01497302250.0090935625

-2.500.02418442440.0175283005

-2.250.0383112950.0317396518

-2.000.05920773160.0539909665

-1.750.08875326720.0862773188

-1.500.12826094490.1295175957

-1.250.17758247320.1826490854

-1.000.23412477290.2419707245

-0.750.2922619750.3011374322

-0.500.34375457030.3520653268

-0.250.37951657760.3866681168

-0.000.39235316280.3989422804

0.250.37951657760.3866681168

0.500.34375457030.3520653268

0.750.2922619750.3011374322

1.000.23412477290.2419707245

1.250.17758247320.1826490854

1.500.12826094490.1295175957

1.750.08875326720.0862773188

2.000.05920773160.0539909665

2.250.0383112950.0317396518

2.500.02418442440.0175283005

2.750.01497302250.0090935625

3.000.00913518390.0044318484

3.250.00551544590.0020290481

3.500.00330731210.0008726827

3.750.00197580220.0003525957

4.000.00117899970.0001338302


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Chi-sq Data


for the Chi-Sq Distribution


P-Value:0.050

Chi-Sq Critical Value:3.841

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Chi-sq Data

3.96952547450

0.70413065350.0298195265

0.4393912894

0.3166058997

0.2419707245

0.1909945646

0.1538663228

0.1257139203

0.1037768743

0.0513934433

0.0269954833

0.0146449826

0.0081086956

0.0045533429

0.0025833732

0.0014772828

0.0008500367

0.0004915799

0.0002854648

0.0001663506

0.0000972265

0.0000569713

0.0000334576

0.0000196871

0.0000116044

0.0000068507

0.00000405

0.0000023972

0.0000014206

0.0000008427

0.0000005003

0.0000002973

0.0000001768

0.0000001053

0.0000000627

0.0000000374

0.0000000223

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F DF

0.013.969525474513.8414553380

0.250.70413065353.8414553380.0298195265

0.500.4393912894

0.750.3166058997

1.000.2419707245

1.250.1909945646

1.500.1538663228

1.750.1257139203

2.000.1037768743

3.000.0513934433

4.000.0269954833

5.000.0146449826

6.000.0081086956

7.000.0045533429

8.000.0025833732

9.000.0014772828

10.000.0008500367

11.000.0004915799

12.000.0002854648

13.000.0001663506

14.000.0000972265

15.000.0000569713

16.000.0000334576

17.000.0000196871

18.000.0000116044

19.000.0000068507

20.000.00000405

21.000.0000023972

22.000.0000014206

23.000.0000008427

24.000.0000005003

25.000.0000002973

26.000.0000001768

27.000.0000001053

28.000.0000000627

29.000.0000000374

30.000.0000000223

18.3070290368


Help

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Page &P

F Data


for the F Distribution

Numerator Degrees of Freedom4

Denominator Degrees of Freedom9

P-Value:0.050

F Critical Value:3.633

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F Data

0.0474997890

0.36851522060.0343745398

0.5621432266

0.6501395955

0.6750848247

0.663290885

0.6310389919

0.5883712874

0.5414240785

0.4938788686

0.4478728006

0.4045740045

0.3645471127

0.3279862159

0.294863456

0.2650236028

0.2382438923

0.2142714629

0.1928463347

0.1737150746

0.156638489

0.1413955232

0.1277847843

0.1156246131

0.1047523037

0.0950228563

0.0863075089

0.0784922007

0.0714760606

0.0651699734

0.0594952532

0.0543824345

0.0497701842

0.0456043305

0.0418370006

0.0384258593

0.0353334387

0.0325265485

0.0299757595

0.0276549507

0.0255409127

0.0236130005

0.0218528296

0.0202440099

0.0187719123

0.0174234649

0.0161869734

0.0150519637

0.0140090444

0.0130497846

0.0121666075

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Smooth1 Data

0.010.04749978943.63309027310

0.100.368515220693.63309027310.0343745398

0.200.5621432266

0.300.6501395955

0.400.6750848247

0.500.663290885

0.600.6310389919

0.700.5883712874

0.800.5414240785

0.900.4938788686

1.000.4478728006

1.100.4045740045

1.200.3645471127

1.300.3279862159

1.400.294863456

1.500.2650236028

1.600.2382438923

1.700.2142714629

1.800.1928463347

1.900.1737150746

2.000.156638489

2.100.1413955232

2.200.1277847843

2.300.1156246131

2.400.1047523037

2.500.0950228563

2.600.0863075089

2.700.0784922007

2.800.0714760606

2.900.0651699734

3.000.0594952532

3.100.0543824345

3.200.0497701842

3.300.0456043305

3.400.0418370006

3.500.0384258593

3.600.0353334387

3.700.0325265485

3.800.0299757595

3.900.0276549507

4.000.0255409127

4.100.0236130005

4.200.0218528296

4.300.0202440099

4.400.0187719123

4.500.0174234649

4.600.0161869734

4.700.0150519637

4.800.0140090444

4.900.0130497846

5.000.0121666075


Help

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Smooth1

ObsYearMonthPercent DiffForecastThetaTheta NormDiff Squared0.18

11981Feb2.8832.8830.00000000010.0000000015025.2234841628

21981Mar3.0052.883021736100.00000000030.0149757534

31981Apr-0.6102.905049319100.000000000412.3530458635

41981May-0.6012.272405117300.00000000058.2584859718

51981Jun-1.4991.755128648200.000000000610.5917601551

61981Jul-2.5121.169318927400.000000000713.5526927083

71981Aug-7.4420.506667251400.000000000963.1760649312

81981Sep-3.572-0.924033452400.00000000117.0140651324

91981Oct0.302-1.400746899100.00000000142.900573299

101981Nov4.273-1.09418764730.00000000010.000000001728.8073812655

111981Dec-1.573-0.12808250140.00000000010.0000000022.0865984747

121981Jan-0.446-0.38809363690.00000000010.00000000250.0033201392

131982Feb-5.362-0.39846535360.00000000010.00000000324.6385200954

141982Mar-0.197-1.29193502360.00000000010.00000000371.199958318

151982Apr3.110-1.09475832740.00000000020.000000004517.6818865907

161982May-3.397-0.33786128980.00000000020.00000000549.3592028839

171982Jun-0.929-0.88853194770.00000000020.00000000660.0016030204

181982Jul-0.410-0.89573874050.00000000030.00000000810.235812082

191982Aug11.465-0.80832986530.00000000040.0000000099150.6467993811

201982Sep-0.5611.40095877580.00000000040.0000000123.8508718475

211982Oct10.6521.04773328430.00000000050.000000014792.2450468292

221982Nov4.7962.77653041450.00000000060.00000001794.0770800042

231982Dec0.6993.13998246380.00000000080.00000002185.9605410005

241983Jan2.7862.70052651780.0000000010.00000002660.0073612868

251983Feb3.4322.71597014740.00000000120.00000003250.5129618449

261983Mar1.5652.84488858580.00000000140.00000003961.6386907017

271983Apr8.5102.61446814680.00000000170.000000048334.7619361174

281983May-2.1383.67573471560.00000000210.000000058933.8031565237

291983Jun1.8322.62920603880.00000000260.00000007180.6360203561

301983Jul-1.8612.48565444690.00000000320.000000087518.8929242464

311983Aug1.4131.70326659840.00000000380.00000010680.0844958807

321983Sep1.3951.65094388310.00000000470.00000013020.065315143

331983Oct-0.6431.60494159370.00000000570.00000015885.053596563

341983Nov4.1481.20029788790.0000000070.00000019368.6883241452

351983Dec-1.3621.73086522790.00000000850.00000023619.5661098649

361984Jan-3.0241.17414091580.00000001040.00000028817.6235375516

371984Feb-5.4030.41849376490.00000001260.000000351233.8917571767

381984Mar0.889-0.62940553010.00000001540.00000042832.3043296985

391984Apr0.503-0.35616519220.00000001880.00000052230.7382538231

401984May-5.629-0.20150613540.00000002290.000000636929.456282668

411984Jun2.494-1.17843169980.0000000280.000000776713.4834581286

421984Jul-1.512-0.51747478490.00000003410.00000094720.9887488057

431984Aug9.782-0.6964593130.00000004160.0000011551109.8043301441

441984Sep-1.4431.18971679130.00000005070.00000140876.9321423811

451984Oct0.0560.7157954760.00000006180.00000171790.4359599169

461984Nov-1.5270.59694640650.00000007540.00000209514.5123120466

471984Dec1.9030.21458674560.0000000920.00000255492.8520095394

481985Jan6.2070.51856883240.00000011220.000003115832.3562303852

491985Feb-0.2141.5424545020.00000013680.00000379973.086855902

501985Mar-1.3421.22620439330.00000016680.00000463386.5951080975

511985Apr-0.6880.76394742760.00000020340.0000056512.1091953033

521985May4.5590.50253218740.00000024810.000006891516.451735648

531985Jun1.5241.23262549310.00000030260.00000840420.0850387819

541985Jul0.8981.28511603060.0000003690.00001024910.1499997798

551985Aug-0.9971.21540238160.000000450.00001249894.8966696715

561985Sep-0.4030.81709082340.00000054870.00001524251.4893423577

571985Oct3.4380.59742131620.00000066920.00001858848.0696144433

581985Nov7.1181.10874851870.00000081610.000022668836.1081425043

591985Dec5.0632.19036943990.00000099520.00002764498.2543707905

601986Jan1.5722.70751701420.00000121370.00003371331.2884670158

611986Feb8.7892.50319782440.00000148010.000041113839.507859175

621986Mar6.4103.63459282050.0000018050.00005013877.7026427222

631986Apr-1.9044.13415825590.00000220120.000061144836.4617892725

641986May5.1983.04725348890.00000268440.00007456684.6254018328

651986Jun0.8533.43437494190.00000327370.00009093526.6630389219

661986Jul-6.2032.96974340940.00000399230.000110896584.1436593988

671986Aug6.9301.31860605450.00000486860.000135239731.4883849206

681986Sep-6.8882.32866727620.00000593740.000164926584.9492286672

691986Oct6.2360.66964497110.00000724070.000201129830.9866457693

701986Nov1.9391.6716266690.00000883010.00024528030.0717525567

711986Dec-0.9551.71984267080.00001076840.00029912237.1545327716

721987Jan13.8241.23837942010.00001313220.0003647833158.3897097563

731987Feb3.0563.50373294570.00001601490.00044485770.2004524421

741987Mar3.6293.42314349770.00001953030.00054250930.0422178964

751987Apr-0.7953.4601280870.00002381750.000661596718.1089637909

761987May0.2282.69414477980.00002904570.00080682536.0824958714

771987Jun5.5402.25021588210.00003542160.000983933310.8246957966

781987Jul6.3482.84243218030.00004319710.001199918712.2924022371

791987Aug3.5333.47352158160.00005267940.00146331540.0035783449

801987Sep-2.5043.48428904990.00006424310.00178453135.8549878764

811987Oct-23.2162.40646642670.00007834530.0021762573656.5061181925

821987Nov-8.025-2.2055611350.0000955430.002653972433.8654118563

831987Dec5.742-3.25305306530.00011651590.003236551780.9085846479

841988Jan1.000-1.63396747690.00014209250.00394701426.9382465789

851988Feb5.791-1.15983754840.00017328360.00481343248.3137733271

861988Mar-4.0340.09130842870.00021132140.00587003917.0145253482

871988Apr2.227-0.65116757860.0002577090.00715858418.282662617

881988May-0.060-0.13313450160.00031427930.00872998060.0054165076

891988Jun5.445-0.11988705480.00038326740.010646317930.9655109185

901988Jul-0.6060.88175287730.00046739930.01298331452.213580541

911988Aug-4.5600.61394695970.00056999920.015833310326.7745456457

921988Sep4.000-0.31744730640.00069512090.01930891518.6378012179

931988Oct1.6920.459640050.00084770850.02354745731.5174937859

941988Nov-1.5890.68137592590.00103379080.02871641145.1541739435

951988Dec2.5570.27272542020.00126072050.03502001395.2161775704

961989Jan8.0120.68382656750.0015374640.04270733453.7049452825

971989Feb-3.5832.00293241130.00187495610.052082114631.2048704324

981989Mar0.8460.9974286560.00228653190.0635147740.0230108936

991989Apr6.2050.97012384040.00278845350.077457041427.4001338967

1001989May2.5361.91233631020.0034005530.09445980660.3894326197

1011989Jun-1.6162.02466447620.00414701590.115194886113.2576016806

1021989Jul9.0411.36926666230.00505733650.140481568458.8518298649

1031989Aug2.8792.75013569580.00616748350.17131898590.0166991172

1041989Sep-1.6242.77339620740.00752132130.208925592519.3360444453

1051989Oct-1.7731.9818863590.00917234310.254787307914.0981401175

1061989Nov2.3131.30603153750.01118578430.31071622921.0146936702

1071989Dec1.7341.48734914530.01364120030.37892223070.0608964142

1081990Jan-5.9081.53176810320.01663561010.462100281455.3506596107

1091990Feb1.4170.19260367290.02028732940.56353692851.4993399849

1101990Mar3.0430.41300924340.02474064560.68724015676.9194107835

1111990Apr-1.8640.8864951560.03017151910.8380977527.5627052981

1121990May8.2770.39148843710.03679453551.022070429362.1812776487

1131990Jun0.1401.81088034770.04487138471.24642735282.7915302793

1141990Jul0.8511.51013862960.05472120091.52003335710.4346775573

1151990Aug-10.0111.39146448390.06673317181.853699216130.0165322727

1161990Sep-6.192-0.6609817750.08138191682.260608830.5916670417

1171990Oct-0.414-1.65655699180.099246242.756841.544278963

1181990Nov4.804-1.43287275360.1210323.36238.8937129926

1191990Dec2.891-0.31030591770.18510.2509908057

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Smooth2

Exponential Smoothing (1 Parameter)

Smoothing Factor0.180Mean Square Error25.223

Final Value-0.310Standard Error5.022

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Smooth2

2.88302173612.883021736100.0000000015

3.00539719672.883021736100.0000000003

-0.60964069052.905049319100.0000000004

-0.60135304432.272405117300.0000000005

-1.49936980091.755128648200.0000000006

-2.51207927281.169318927400.0000000007

-7.44166999180.506667251400.0000000009

-3.5724414898-0.924033452400.0000000011

0.3023600555-1.400746899100.0000000014

4.2730631635-1.094187647300.0000000017

-1.5725888096-0.128082501400.000000002

-0.4457142857-0.388093636900.0000000025

-5.3621857422-0.398465353600.000000003

-0.1965089339-1.291935023600.0000000037

3.1102252148-1.094758327400.0000000045

-3.3971427224-0.337861289800.0000000054

-0.9285696854-0.888531947700.0000000066

-0.4101338785-0.895738740500.0000000081

11.4654959189-0.808329865300.0000000099

-0.5614050661.400958775800.000000012

10.65216178521.047733284300.0000000147

4.79570846612.776530414500.0000000179

0.69856054193.139982463800.0000000218

2.78632445962.700526517800.0000000266

3.43218369432.715970147400.0000000325

1.56477503552.844888585800.0000000396

8.51039352942.614468146800.0000000483

-2.13831348883.675734715600.0000000589

1.8316971952.629206038800.0000000718

-1.86094471182.485654446900.0000000875

1.41258484681.703266598400.0000001068

1.39537560851.650943883100.0000001302

-0.64307899411.604941593700.0000001588

4.14789422141.200297887900.0000001936

-1.36204761681.730865227900.0000002361

-3.02389881141.174140915800.000000288

-5.40316898520.418493764900.0000003512

0.8885963469-0.629405530100.0000004283

0.5030517903-0.356165192200.0000005223

-5.6288703822-0.201506135400.0000006369

2.4935511608-1.178431699800.0000007767

-1.5118332745-0.517474784900.0000009472

9.7822968223-0.69645931300.0000011551

-1.44317940511.189716791300.0000014087

0.0555228680.71579547600.0000017179

-1.5272739320.596946406500.0000020951

1.90337611650.214586745600.0000025549

6.20682255260.518568832400.0000031158

-0.21449054611.54245450200.0000037997

-1.34188986071.226204393300.0000046338

-0.68835946260.763947427600.000005651

4.55860610780.502532187400.0000068915

1.52423959071.232625493100.0000084042

0.89781798031.285116030600.0000102491

-0.99743960811.215402381600.0000124989

-0.40329532760.817090823400.0000152425

3.43812799650.597421316200.0000185884

7.11775363641.108748518700.0000226688

5.06341151942.190369439900.0000276449

1.57241040432.707517014200.0000337133

8.78872558072.503197824400.0000411138

6.40995635033.634592820500.0000501387

-1.90420156054.134158255900.0000611448

5.19792822793.047253488900.0000745668

0.85308864983.434374941900.0000909352

-6.20324189532.969743409400.0001108965

6.93005728581.318606054500.0001352397

-6.88812330772.328667276200.0001649265

6.23620995940.669644971100.0002011298

1.9394933461.67162666900.0002452803

-0.95495316651.719842670800.0002991223

13.82367678471.238379420100.0003647833

3.05601379033.503732945700.0004448577

3.6286134383.423143497700.0005425093

-0.79533473053.46012808700.0006615967

0.22787312582.694144779800.0008068253

5.54030642752.250215882100.0009839333

6.348484412.842432180300.0011999187

3.533340853.473521581600.0014633154

-2.50361441263.484289049900.001784531

-23.21590891582.406466426700.0021762573

-8.024960748-2.20556113500.0026539724

5.7418668703-3.253053065300.0032365517

1.0000876817-1.633967476900.0039470142

5.7909734351-1.159837548400.004813432

-4.03355827810.091308428700.005870039

2.2267939599-0.651167578600.0071585841

-0.0595375751-0.133134501600.0087299806

5.4447792351-0.119887054800.0106463179

-0.60605777630.881752877300.0129833145

-4.56046562970.613946959700.0158333103

3.9997046735-0.317447306400.019308915

1.69150602720.4596400500.0235474573

-1.58890466110.681375925900.0287164114

2.55662068280.272725420200.0350200139

8.01219236640.683826567500.042707334

-3.58319956282.002932411300.0520821146

0.84573523620.99742865600.063514774

6.20463756150.970123840400.0774570414

2.53638167691.912336310200.0944598066

-1.61643448982.024664476200.1151948861

9.04076129281.369266662300.1404815684

2.87936076012.750135695800.1713189859

-1.62388072792.773396207400.2089255925

-1.77286264961.98188635900.2547873079

2.3133515811.306031537500.3107162292

1.73412113351.487349145300.3789222307

-5.90803428741.531768103200.4621002814

1.41707906460.192603672900.5635369285

3.04348653540.413009243400.6872401567

-1.86354217070.88649515600.838097752

8.27699905150.391488437101.0220704293

0.14009302451.810880347701.2464273528

0.85083782011.510138629601.5200333571

-10.01101473221.391464483901.853699216

-6.1919552013-0.66098177502.2606088

-0.4138667797-1.656556991802.75684

4.8036096678-1.432872753603.362

2.8914109351-0.310305917705

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Weights for Smoothing Factor

Observed and Forecasted

Percent Diff

Forecast

Month

Theta Norm

Smooth2 Data

Exponential Smoothing (2 Parameters)

Smoothing FactorsMean Square Error31,035.85

Location0.15Standard Error176.17

Trend0.40Final Observation2,633.66

Forecasted Trend-7.36


Help

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Page &P

Smooth2 Data

636.46146831960947.2700

698.36568396090974.5800

750.606192900401003.870

796.83818088550997.750

831.55320390470991.750

857.96967341030976.880

876.34619245350952.340

886.59684561830881.470

880.88656799530849.980

871.39643232780852.550

865.81117719770888.980

869.817326449508750

870.0744229810871.10

869.68207683320824.390

859.75795568780822.770

852.23437656240848.360

852.00338861470819.540

845.61478144450811.930

839.1940049010808.60

833.58426856980901.310

848.69334692660896.250

858.49337607310991.720

885.805488373301039.280

916.077136344101046.540

940.432448628501075.70

965.343710975901112.620

992.423032114501130.030

1017.101403968301226.20

1056.658789375601199.980

1081.499328570801221.960

1105.892643846201199.220

1120.521876005801216.160

1136.262071846901233.130

1152.64714741501225.20

1164.167207109801276.020

1184.376605127601258.640

1196.53220180901220.580

1198.46542400801154.630

1186.92874188901164.890

1181.26170949501170.750

1178.619820404301104.850

1162.961267743801132.40

1157.274229822401115.280

1149.28618669301224.380

1166.082913495501206.710

1174.436369355101207.380

1180.773749682101188.940

1181.72978866801211.570

1187.402281230701286.770

1207.537315563801284.010

1221.980544139901266.780

1229.450058065301258.060

1233.668106683401315.410

1249.326325077601335.460

1265.487536953901347.450

1280.446002793701334.010

1289.272660204301328.630

1295.328695871501374.310

1310.008152309401472.130

1341.894985954201546.670

1381.624771955801570.990

1416.896876499301709.060.0000000001

1473.125937226501818.610.0000000001

1538.964301963801783.980.0000000002

1582.04830986201876.710.0000000004

1635.610355298301892.720.0000000006

1680.75507715301775.310.000000001

1691.840680669701898.340.0000000017

1728.33879762301767.580.0000000029

1729.693309411801877.810.0000000048

1755.805611859401914.230.0000000081

1784.091349396101895.950.0000000135

1802.486594576102158.040.0000000225

1872.514873943302223.990.0000000374

1939.21030398402304.690.0000000624

2007.258755065702286.360.0000001039

2055.960039813402291.570.0000001732

2095.715792645902418.530.0000002887

2154.404079031802572.070.0000004812

2232.426060346602662.950.000000802

2311.680707196402596.280.0000013367

2359.17603465801993.530.0000022279

2271.04738958401833.550.0000037132

2175.854937221201938.830.0000061887

2128.877290282901958.220.0000103144

2097.587752927402071.620.0000171907

2097.082023700401988.060.0000286512

2077.748780372602032.330.000047752

2071.804199453102031.120.0000795866

2066.14991121702141.710.0001326444

2084.388329543502128.730.0002210739

2094.403423117402031.650.0003684565

2081.183915453102112.910.0006140942

2088.534531617102148.650.0010234904

2101.251273963702114.510.0017058173

2103.411336122702168.570.0028430288

2116.531487656402342.320.0047383813

2162.832575679202258.390.0078973022

2179.590076030102277.490.0131621704

2196.875396647202418.80.0219369506

2240.202886363102480.150.0365615844

2285.899232936402440.060.060935974

2312.440515712702660.660.1015599567

2379.959738985802737.270.1692665944

2447.324658515902692.820.2821109907

2489.742987965302645.080.4701849846

2513.535556706602706.270.783641641

2546.412333362302753.21.3060694016

2582.626249454902590.542.176782336

2577.171171694902627.253.62797056

2582.900311735602707.216.0466176

2605.274754452202656.7610.077696

2612.207334970502876.6616.79616

2664.595110936602880.6927.9936

2704.883070023202905.246.656

2741.125444955902614.3677.76

2708.377129474202452.48129.6

2652.039660810102442.33216

2609.202472142802559.65360

2601.98035759402633.66600

2611.29935771841000

2603.938338133Feb1991

2596.5773185477Mar1991

2589.2162989624Apr1991

2581.855279377May1991

2574.4942597917Jun1991

2567.1332402063Jul1991

2559.772220621Aug1991

2552.4112010357Sep1991

2545.0501814503Oct1991

2537.689161865Nov1991

2530.3281422796Dec1991

2522.9671226943Jan1992

2515.606103109Feb1992

2508.2450835236Mar1992

2500.8840639383Apr1992

2493.5230443529May1992

2486.1620247676Jun1992

2478.8010051822Jul1992

2471.4399855969Aug1992

2464.0789660116Sep1992

2456.7179464262Oct1992

2449.3569268409Nov1992

2441.9959072556Dec1992

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Weights for the Trend Factor

Observed and Forecasted Dow Jones Averages

Forecast

Month

Dow Jones

ObsYearMonthDow JonesForecastS(t)T(t)Diff SquaredParameters

11981Jan947.27636.4614683196618.747910364117.713557955496601.9433653516000.150.40

21981Feb974.58698.3656839609668.026223809530.339460151476294.34838493500618.747910364117.7135579554

31981Mar1003.87750.6061929004714.009290238136.596902662364142.555986601900

41981Apr997.75796.8381808855757.488396702439.349784183140365.559059915200

51981May991.75831.5532039047793.52763719738.025566707725663.013479190800

61981Jun976.88857.9696734103823.260991617534.708681792814139.665769676200

71981Jul952.34876.3461924535846.303842874830.04234957865775.058785417300

81981Aug881.47886.5968456183862.209266443624.387579174726.28454599400

91981Sep849.98880.8865679953865.098376477115.7881915182955.215945246500

101981Oct852.55871.3964323278862.83062000558.5658123223355.188011486700

111981Nov888.98865.8111771977861.28852700474.522650193536.794350042800

121981Dec875.00869.8173264495865.4422479544.375078495526.860105130700

131982Jan871.10870.074422981866.87591076093.19851222011.05180822200

141982Feb824.39869.6820768332867.50952414682.17255268642051.372223860400

151982Mar822.77859.7579556878861.0415955247-1.2836398371368.108865960900

161982Apr848.36852.2343765624855.300856196-3.066479633715.01079374700

171982May819.54852.0033886147854.2597277666-2.2563391521053.871600346800

181982Jun811.93845.6147814445849.0517686016-3.43698715721134.664500960900

191982Jul808.60839.194004901843.4835033114-4.2894984104935.993135881500

201982Aug901.31833.5842685698838.2509778147-4.66670924494586.774697761800

211982Sep896.25848.6933469266847.70983114250.98351578422261.635251539800

221982Oct991.72858.4933760731854.99085647113.50251960217749.333322971500

231982Nov1039.28885.8054883733875.500228000410.305260372923554.425719040800

241982Dec1046.54916.0771363441900.067193800416.009942543717020.558793300700

251983Jan1075.70940.4324486285922.038114730318.394333898218297.310454034700

261983Feb1112.62965.3437109759945.087397520820.256313455121690.305308717500

271983Mar1130.03992.4230321145970.217287892722.205744221818935.677610648900

281983Apr1226.201017.1014039683994.189194708822.912209259543722.222862433500

291983May1199.981056.65878937561028.990815502427.667973873220540.969414833300

301983Jun1221.961081.49932857081054.639193177126.860135393819729.200218328800

311983Jul1199.221105.89264384621079.737314200526.15532964568709.995406667200

321983Aug1216.161120.52187600581097.659717070422.86215893549146.650761132100

331983Sep1233.131136.26207184691115.434759509920.8273123379383.395504682500

341983Oct1225.201152.6471474151133.089045583419.55810183165263.916418222200

351983Nov1276.021164.16720710981146.905688745917.26151836412511.047277329400

361983Dec1258.641184.37660512761166.27283543418.10376969365515.05181797100

371984Jan1220.581196.5322018091180.127910118916.4042916901578.296597833500

381984Feb1154.631198.4654240081186.195723601112.26970040691921.544397962100

391984Mar1164.891186.9287418891181.46086506095.4678768281485.706144050300

401984Apr1170.751181.2617094951178.97523530182.2864741932110.496036506800

411984May1104.851178.61982040431177.74145000650.87837039785441.98640248600

421984Jun1132.401162.96126774381166.8077325055-3.8464647617933.9910861100

431984Jul1115.281157.27422982241161.6465726297-4.37234280741763.515338372200

441984Aug1224.381149.2861866931154.6915867352-5.40540004225639.080796978900

451984Sep1206.711166.08291349551165.1448487250.93806477061650.560157841900

461984Oct1207.381174.43636935511171.37962141623.05674793881085.282800070400

471984Nov1188.941180.77374968211176.77967820383.994071478366.687644254300

481984Dec1211.571181.7297886681178.60372647323.1260621948890.438212339100

491985Jan1286.771187.40228123071183.54866750223.85361372859873.94353341400

501985Feb1284.011207.53731556381199.03186737698.50544818695848.071464871600

511985Mar1266.781221.98054413991211.778587270410.20195686962006.991245359400

521985Apr1258.061229.45005806531220.02879917989.4212588855818.528777505800

531985May1315.411233.66810668341225.73347930287.93462738056681.737122989700

541985Jun1335.461249.32632507761239.184957407410.14136767017419.009955645900

551985Jul1347.451265.48753695391253.626213796311.86132315766717.845348576800

561985Aug1334.011280.44600279371267.699781726812.74622106682869.101796720700

571985Sep1328.631289.27266020431277.646314467811.62634573651549.000195794300

581985Oct1374.311295.32869587151285.293867297610.03482857386238.04640184400

591985Nov1472.131310.00815230941298.64628720311.361865106426283.493498603200

601985Dec1546.671341.89498595421324.668844122617.226141831741932.806377444800

611986Jan1570.991381.62477195581357.969017504223.655754451735859.189592222700

621986Feb1709.061416.89687649931389.922164878626.974711620785359.290733692900.0000000001

631986Mar1818.611473.12593722651437.792840146835.3330970797119359.23763048700.0000000001

641986Apr1783.981538.96430196381494.915414124844.04888783960032.69228417500.0000000002

651986May1876.711582.0483098621538.27510200643.773207855986825.511634999500.0000000004

661986Jun1892.721635.61035529831589.040336705146.570018593266105.369398619100.0000000006

671986Jul1775.311680.7550771531634.592286199446.16279095368940.633434607300.000000001

681986Aug1898.341691.84068066971655.699943269536.140737400242641.968883892800.0000000017

691986Sep1767.581728.3387976231692.09595177936.2428458441539.871963993100.0000000029

701986Oct1877.811729.69330941181703.418559012226.274750399621938.554030796500.0000000048

711986Nov1914.231755.80561185941729.577275160426.22833669925098.286757722600.0000000081

721986Dec1895.951784.09134939611757.275183886326.816165509812512.357714923400.0000000135

731987Jan2158.041802.48659457611778.076406303424.4101882727126418.22410855900.0000000225

741987Feb2223.991872.51487394331835.070945357837.4439285854123534.7642365900.0000000374

751987Mar2304.691939.2103039841893.408803554245.8015004298133575.40819997900.0000000624

761987Apr2286.362007.25875506571955.10098302152.157772044677897.504923894300.0000001039

771987May2291.572055.96003981342004.789835567951.170204245555512.05333913160.00000000010.0000001732

781987Jun2418.532095.71579264592047.806860232747.9089324132104209.0124696310.00000000010.0000002887

791987Jul2572.072154.40407903182103.415331197850.988747834174444.8215382330.00000000020.0000004812

801987Aug2662.952232.42606034662173.713531518158.7125288285185350.8626146420.00000000030.000000802

811987Sep2596.282311.68070719642247.099001790464.58170540680996.75746429030.00000000050.0000013367

821987Oct1993.532359.1760346582299.476151521859.6998831362133697.0226611470.00000000090.0000022279

831987Nov1833.552271.0473895842253.584228793617.4631607904191403.9658927910.00000000150.0000037132

841987Dec1938.832175.85493722122190.5790944745-14.724157253456180.82086469620.00000000250.0000061887

851988Jan1958.222128.87729028292152.8167303034-23.939440020529123.91072668820.00000000410.0000103144

861988Feb2071.622097.58775292742123.6272207578-26.0394678305674.32419209610.00000000690.0000171907

871988Mar1988.062097.08202370042115.8261376442-18.744113943811885.80165173140.00000001150.0000286512

881988Apr2032.332077.74878037262096.6612169976-18.91243662492062.86561053760.00000001910.000047752

891988May2031.122071.80419945312087.0115344479-15.20733499481655.20408514080.00000003180.0000795866

901988Jun2141.712066.1499112172078.6278042807-12.47789306385709.3270168990.00000005310.0001326444

911988Jul2128.732084.38832954352088.0901336386-3.70180409511966.18373887090.00000008840.0002210739

921988Aug2031.652094.40342311742094.18611359280.21730952463937.99211295770.00000014740.0003684565

931988Sep2112.912081.18391545312084.8056965539-3.62178110081006.54444067660.00000024560.0006140942

941988Oct2148.652088.53453161712089.0213420708-0.48681045373613.86953889480.00000040940.0010234904

951988Nov2114.512101.25127396372097.96564076023.2856332035175.79381610520.00000068230.0017058173

961988Dec2168.572103.41133612272100.44729464622.96404147654245.65147827920.00000113720.0028430288

971989Jan2342.322116.53148765642110.66570044925.865787207150980.45230634890.00000189540.0047383813

981989Feb2258.392162.83257567922145.413845381817.41873029739131.22134282950.00000315890.0078973022

991989Mar2277.492179.59007603012162.360268574617.22980745559584.39511332010.00000526490.0131621704

1001989Apr2418.802196.87539664722179.629728288417.245668358849250.52957329550.00000877480.0219369506

1011989May2480.152240.20288636312215.505269045124.69761731857574.61734267140.00001462460.0365615844

1021989Jun2440.062285.89923293642255.201978688430.697254248123765.54210162380.00002437440.060935974

1031989Jul2660.662312.44051571272282.930681885129.5098338275121256.8092373440.0000406240.1015599567

1041989Aug2737.272379.95973898582339.590079602340.3696593834127670.6226260620.00006770660.1692665944

1051989Sep2692.822447.32465851592399.24206766248.082590853960267.96269039360.00011284440.2821109907

1061989Oct2645.082489.74298796532443.278757512746.464230452624129.58730786230.0001880740.4701849846

1071989Nov2706.272513.53555670662473.548943885839.986612820837146.56563161560.00031345670.783641641

1081989Dec2753.202546.41233336232508.457102302937.955231059342761.13907348040.00052242781.3060694016

1091990Jan2590.542582.62624945492545.168536957537.457712497462.62744768990.00087071292.176782336

1101990Feb2627.252577.17117169492551.974256413925.1969152812507.88904441480.00145118823.62797056

1111990Mar2707.212582.90031173562563.265617951819.634693783815452.89859640360.0024186476.0466176

1121990Apr2656.762605.27475445222584.85727525920.41747919322650.73050912030.004031078410.077696

1131990May2876.662612.20733497052595.642683970216.564651000469935.21204119660.00671846416.79616

1141990Jun2880.692664.59511093662637.795281374626.79982956246697.00107930620.0111974427.9936

1151990Jul2905.202704.88307002322674.229489168430.653580854740126.87243534430.018662446.656

1161990Aug2614.362741.12544495592708.875065793232.250379162716069.47803486790.03110477.76

1171990Sep2452.482708.37712947422694.697805924213.6793235565483.34087315830.05184129.6

1181990Oct2442.332652.03966081012658.3651350356-6.325474225443978.14183710570.0864216

1191990Nov2559.652609.20247214282625.9598647802-16.75739263742455.44749546740.144360

1201990Dec2633.662601.9803575942616.0133850632-14.03302746921003.59974297510.24600

1211991Jan2611.29935771842618.6603773037-7.36101958530.401000

1221991Feb2603.93833813331035.8460596471

1231991Mar2596.5773185477

1241991Apr2589.2162989624

1251991May2581.855279377

1261991Jun2574.4942597917

1271991Jul2567.1332402063

1281991Aug2559.772220621

1291991Sep2552.4112010357

1301991Oct2545.0501814503

1311991Nov2537.689161865

1321991Dec2530.3281422796

1331992Jan2522.9671226943

1341992Feb2515.606103109

1351992Mar2508.2450835236

1361992Apr2500.8840639383

1371992May2493.5230443529

1381992Jun2486.1620247676

1391992Jul2478.8010051822

1401992Aug2471.4399855969

1411992Sep2464.0789660116

1421992Oct2456.7179464262

1431992Nov2449.3569268409

1441992Dec2441.9959072556


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Confidence Interval for Proportion-IGeneral Case: Confidence Interval (CI) for level (1-)*100 % (e.g. = 0.05 corresponds to 95% level)Then the CI (1-) is given by p - z (1-/2) p p + z (1-/2) p, where z (1-/2) is the z-score for the (1-/2) centile, is the population proportion (proportion yeses in a yes/no questionnaire, proportion test positive in a medical test, etc.), p = x /N is the sample estimate of (x is the number of successes in a sample size N) and p, the standard deviation of the proportion, is estimated by p = {p(1-p)/N} 1/2

Confidence Interval for Proportion-IIThe CI (1-) for 1- = 0.95, a 95% CI, is given by p - 1.96 p p + 1.96 p, p = {p(1-p)/N} 1/2Example (Ex. 7.20, text): 84 out of 125 individuals are aware of a certain product; a 95% CI for this proportion is given by p = 84/125 = 0.672; 1.96 p = 1.96 x [p(1-p)/N] = 0.082, so 95% CI is given by 0.672 - 0.082 to 0.672 + 0.082 or (0.590, 0.754)

Sample Size Required for Given CI widthWe know that the CI gets smaller as the sample size, N, increases. Suppose we require (at a certain significance level) a specific width, E, for the CI.Then E = 2 z (1-/2) mean and, since mean = /N, we get N = (2 z (1-/2) /E)2Example: Exercise 7.25, text: want 95% CI for insurance claims to $50 wide (=E), with estimated $400;Then N = (2x1.96x400 /50)2 = 984.

Students t-Distribution for Unknown Sample Standard Deviation, s, used to estimate population standard deviation, , if unknowns = { (xi - xbar)2/(N-1)}(1/2) for sample, size N, with mean xbarUncertainty in standard deviation (from sample size estimate) is clearly bigger, the smaller the sample size;Have to account for this uncertainty by use of a new statistic, the Students-t variable.t = (x - ) / s, for individual value of sample, ort= (xbar - ) / SEM, with SEM = s /N, for sample mean.

Students t-Distribution--Continued I The Students t-distribution gives the probability of the t-statistic (see previous slide) occurring by chanceThe distribution will clearly depend on sample size, NThe larger the sample size, the more nearly the sample standard deviation, s, should approach the value for population standard deviation, The effective sample size is the degrees of freedom (abbreviated as df); df = N-1Probability for large t, small N, is lower than for same value of z (see Concepts illustration).

Students-t DistributionThe graph at left compares a Students-t distribution (solid blue line) with the standard normal bell curve (dotted red line) for df=2 (N=3). Note that the probability of the Students-t is less than that for the z-curve, for statistic values greater than 2, or less than -2.

Chart1

0.01872411090.0001338302

0.02113260650.0003525957

0.02402338760.0008726827

0.02752950370.0020290481

0.03183098860.0044318484

0.03717487720.0090935625

0.04390481190.0175283005

0.05250472350.0317396518

0.06366197720.0539909665

0.07835320270.0862773188

0.09794150340.1295175957

0.12421849220.1826490854

0.15915494310.2419707245

0.20371832710.3011374322

0.25464790890.3520653268

0.29958577520.3866681168

0.31830988610.3989422804

0.29958577520.3866681168

0.25464790890.3520653268

0.20371832710.3011374322

0.15915494310.2419707245

0.12421849220.1826490854

0.09794150340.1295175957

0.07835320270.0862773188

0.06366197720.0539909665

0.05250472350.0317396518

0.04390481190.0175283005

0.03717487720.0090935625

0.03183098860.0044318484

0.02752950370.0020290481

0.02402338760.0008726827

0.02113260650.0003525957

0.01872411090.0001338302

Standard Normal

t Distribution

Title Page

Statistical Concepts

for Excel 97

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Box Plots



Central LimitTheorem

ConfidenceIntervals

HypothesisTesting

t Distribution

Chi-sq Distribution

F Distribution

TESTS AND SIMULATIONS

DISTRIBUTIONS

TIME SERIES



Box Plot

DataBox PlotsXOuter Y1Inner Y1Stem X1XBBox1XFQ0

00.0000.0000.0000.00.0001.50

Box Plot Parameters0.00.000

VariableData0.00.000XTQ0

Outer Upper Fence0.0000.00.0001.50

Inner Upper Fence0.0000.00.000

Upper Whisker0.0000.00.000XM0

Maximum0.0000.00.0001.50

Third Quartile0.0000.00.000

Median0.0000.00.000XLW0

First Quartile0.0000.00.0001.00

Minimum0.0000.00.000

Lower Whisker0.0000.00.000XUW0

Inner Lower Fence0.0000.00.0001.00

Outer Lower Fence0.0000.00.000

Interquartile Range0.000XILF0

0.50

1.50

XIUF0

0.50

1.50

XOLF0

0.50

1.50

XOUF0

0.50

1.50

Outliers

2.50

2.50

0

2.50

0

2.40

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Box Plot

000000000000000

00000000

0

0

0

0

0

0

0

0

0

0

0

0

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Box1

Inner Y1

Outer Y1

Outliers

Outliers

Density Function


-1.86

1.76

114.00

476.00

Probability that the value is between -1.86 & 1.76 :92.9%

Reset

Box Plot

Help


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Density Function

0.00443184840

0.00514264090

0.00595253240

0.00687276670

0.00791545160

0.00909356250

0.01042093480

0.01191224360

0.01358296920

0.01544934710

0.01752830050

0.01983735440

0.02239453030

0.02521821990

0.02832703770

0.03173965180

0.03547459280

0.03955004160

0.0439835960

0.04879201860

0.05399096650

0.05959470610

0.06561581480

0.07206487430.0720648743

0.07895015830.0789501583

0.08627731880.0862773188

0.09404907740.0940490774

0.10226492460.1022649246

0.11092083470.1109208347

0.12000900070.1200090007

0.12951759570.1295175957

0.13943056640.1394305664

0.14972746560.1497274656

0.16038332730.1603833273

0.1713685920.171368592

0.18264908540.1826490854

0.1941860550.194186055

0.20593626870.2059362687

0.2178521770.217852177

0.22988214070.2298821407

0.24197072450.2419707245

0.25405905650.2540590565

0.26608524990.2660852499

0.27798488610.2779848861

0.28969155280.2896915528

0.30113743220.3011374322

0.31225393340.3122539334

0.32297235970.3229723597

0.33322460290.3332246029

0.3429438550.342943855

0.35206532680.3520653268

0.36052696250.3605269625

0.36827014030.3682701403

0.37524034690.3752403469

0.38138781550.3813878155

0.38666811680.3866681168

0.3910426940.391042694

0.39447933090.3944793309

0.39695254750.3969525475

0.39844391410.3984439141

0.39894228040.3989422804

0.39844391410.3984439141

0.39695254750.3969525475

0.39447933090.3944793309

0.3910426940.391042694

0.38666811680.3866681168

0.38138781550.3813878155

0.37524034690.3752403469

0.36827014030.3682701403

0.36052696250.3605269625

0.35206532680.3520653268

0.3429438550.342943855

0.33322460290.3332246029

0.32297235970.3229723597

0.31225393340.3122539334

0.30113743220.3011374322

0.28969155280.2896915528

0.27798488610.2779848861

0.26608524990.2660852499

0.25405905650.2540590565

0.24197072450.2419707245

0.22988214070.2298821407

0.2178521770.217852177

0.20593626870.2059362687

0.1941860550.194186055

0.18264908540.1826490854

0.1713685920.171368592

0.16038332730.1603833273

0.14972746560.1497274656

0.13943056640.1394305664

0.12951759570.1295175957

0.12000900070.1200090007

0.11092083470.1109208347

0.10226492460.1022649246

0.09404907740.0940490774

0.08627731880.0862773188

0.07895015830

0.07206487430

0.06561581480

0.05959470610

0.05399096650

0.04879201860

0.0439835960

0.03955004160

0.03547459280

0.03173965180

0.02832703770

0.02521821990

0.02239453030

0.01983735440

0.01752830050

0.01544934710

0.01358296920

0.01191224360

0.01042093480

0.00909356250

0.00791545160

0.00687276670

0.00595253240

0.00514264090

0.00443184840

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Distribution Data

-30.00443184840-1.86

-2.950.005142640901.76

-2.90.00595253240

-2.850.00687276670

-2.80.00791545160

-2.750.00909356250

-2.70.01042093480

-2.650.01191224360

-2.60.01358296920

-2.550.01544934710

-2.500.01752830050

-2.450.01983735440

-2.400.02239453030

-2.350.02521821990

-2.300.02832703770

-2.250.03173965180

-2.200.03547459280

-2.150.03955004160

-2.100.0439835960

-2.050.04879201860

-2.000.05399096650

-1.950.05959470610

-1.900.06561581480

-1.850.07206487430.0720648743

-1.800.07895015830.0789501583

-1.750.08627731880.0862773188

-1.700.09404907740.0940490774

-1.650.10226492460.1022649246

-1.600.11092083470.1109208347

-1.550.12000900070.1200090007

-1.500.12951759570.1295175957

-1.450.13943056640.1394305664

-1.400.14972746560.1497274656

-1.350.16038332730.1603833273

-1.300.1713685920.171368592

-1.250.18264908540.1826490854

-1.200.1941860550.194186055

-1.150.20593626870.2059362687

-1.100.2178521770.217852177

-1.050.22988214070.2298821407

-1.000.24197072450.2419707245

-0.950.25405905650.2540590565

-0.900.26608524990.2660852499

-0.850.27798488610.2779848861

-0.800.28969155280.2896915528

-0.750.30113743220.3011374322

-0.700.31225393340.3122539334

-0.650.32297235970.3229723597

-0.600.33322460290.3332246029

-0.550.3429438550.342943855

-0.500.35206532680.3520653268

-0.450.36052696250.3605269625

-0.400.36827014030.3682701403

-0.350.37524034690.3752403469

-0.300.38138781550.3813878155

-0.250.38666811680.3866681168

-0.200.3910426940.391042694

-0.150.39447933090.3944793309

-0.100.39695254750.3969525475

-0.050.39844391410.3984439141

-0.000.39894228040.3989422804

0.050.39844391410.3984439141

0.100.39695254750.3969525475

0.150.39447933090.3944793309

0.200.3910426940.391042694

0.250.38666811680.3866681168

0.300.38138781550.3813878155

0.350.37524034690.3752403469

0.400.36827014030.3682701403

0.450.36052696250.3605269625

0.500.35206532680.3520653268

0.550.3429438550.342943855

0.600.33322460290.3332246029

0.650.32297235970.3229723597

0.700.31225393340.3122539334

0.750.30113743220.3011374322

0.800.28969155280.2896915528

0.850.27798488610.2779848861

0.900.26608524990.2660852499

0.950.25405905650.2540590565

1.000.24197072450.2419707245

1.050.22988214070.2298821407

1.100.2178521770.217852177

1.150.20593626870.2059362687

1.200.1941860550.194186055

1.250.18264908540.1826490854

1.300.1713685920.171368592

1.350.16038332730.1603833273

1.400.14972746560.1497274656

1.450.13943056640.1394305664

1.500.12951759570.1295175957

1.550.12000900070.1200090007

1.600.11092083470.1109208347

1.650.10226492460.1022649246

1.700.09404907740.0940490774

1.750.08627731880.0862773188

1.800.07895015830

1.850.07206487430

1.900.06561581480

1.950.05959470610

2.000.05399096650

2.050.04879201860

2.100.0439835960

2.150.03955004160

2.200.03547459280

2.250.03173965180

2.300.02832703770

2.350.02521821990

2.400.02239453030

2.450.01983735440

2.500.01752830050

2.550.01544934710

2.600.01358296920

2.650.01191224360

2.700.01042093480

2.750.00909356250

2.800.00791545160

2.850.00687276670

2.900.00595253240

2.950.00514264090

3.000.00443184840


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Target


Vertical Deviation 50 (1-tailed)

mu 50 (2-tailed)


Help

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t-Distribution Data



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t-Distribution Data

0.01872411090.0001338302

0.02113260650.0003525957

0.02402338760.0008726827

0.02752950370.0020290481

0.03183098860.0044318484

0.03717487720.0090935625

0.04390481190.0175283005

0.05250472350.0317396518

0.06366197720.0539909665

0.07835320270.0862773188

0.09794150340.1295175957

0.12421849220.1826490854

0.15915494310.2419707245

0.20371832710.3011374322

0.25464790890.3520653268

0.29958577520.3866681168

0.31830988610.3989422804

0.29958577520.3866681168

0.25464790890.3520653268

0.20371832710.3011374322

0.15915494310.2419707245

0.12421849220.1826490854

0.09794150340.1295175957

0.07835320270.0862773188

0.06366197720.0539909665

0.05250472350.0317396518

0.04390481190.0175283005

0.03717487720.0090935625

0.03183098860.0044318484

0.02752950370.0020290481

0.02402338760.0008726827

0.02113260650.0003525957

0.01872411090.0001338302

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Standard Normal

t Distribution

Chi-sq DF

-4.000.01872411090.00013383021

-3.750.02113260650.0003525957

-3.500.02402338760.0008726827

-3.250.02752950370.0020290481

-3.000.03183098860.0044318484

-2.750.03717487720.0090935625

-2.500.04390481190.0175283005

-2.250.05250472350.0317396518

-2.000.06366197720.0539909665

-1.750.07835320270.0862773188

-1.500.09794150340.1295175957

-1.250.12421849220.1826490854

-1.000.15915494310.2419707245

-0.750.20371832710.3011374322

-0.500.25464790890.3520653268

-0.250.29958577520.3866681168

-0.000.31830988610.3989422804

0.250.29958577520.3866681168

0.500.25464790890.3520653268

0.750.20371832710.3011374322

1.000.15915494310.2419707245

1.250.12421849220.1826490854

1.500.09794150340.1295175957

1.750.07835320270.0862773188

2.000.06366197720.0539909665

2.250.05250472350.0317396518

2.500.04390481190.0175283005

2.750.03717487720.0090935625

3.000.03183098860.0044318484

3.250.02752950370.0020290481

3.500.02402338760.0008726827

3.750.02113260650.0003525957

4.000.01872411090.0001338302


Help

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Chi-sq Data


for the Chi-Sq Distribution


P-Value:0.050

Chi-Sq Critical Value:3.841

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Chi-sq Data

3.96952547450

0.70413065350.0298195265

0.4393912894

0.3166058997

0.2419707245

0.1909945646

0.1538663228

0.1257139203

0.1037768743

0.0513934433

0.0269954833

0.0146449826

0.0081086956

0.0045533429

0.0025833732

0.0014772828

0.0008500367

0.0004915799

0.0002854648

0.0001663506

0.0000972265

0.0000569713

0.0000334576

0.0000196871

0.0000116044

0.0000068507

0.00000405

0.0000023972

0.0000014206

0.0000008427

0.0000005003

0.0000002973

0.0000001768

0.0000001053

0.0000000627

0.0000000374

0.0000000223

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F DF

0.013.969525474513.8414553380

0.250.70413065353.8414553380.0298195265

0.500.4393912894

0.750.3166058997

1.000.2419707245

1.250.1909945646

1.500.1538663228

1.750.1257139203

2.000.1037768743

3.000.0513934433

4.000.0269954833

5.000.0146449826

6.000.0081086956

7.000.0045533429

8.000.0025833732

9.000.0014772828

10.000.0008500367

11.000.0004915799

12.000.0002854648

13.000.0001663506

14.000.0000972265

15.000.0000569713

16.000.0000334576

17.000.0000196871

18.000.0000116044

19.000.0000068507

20.000.00000405

21.000.0000023972

22.000.0000014206

23.000.0000008427

24.000.0000005003

25.000.0000002973

26.000.0000001768

27.000.0000001053

28.000.0000000627

29.000.0000000374

30.000.0000000223

18.3070290368


Help

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Page &P

F Data


for the F Distribution

Numerator Degrees of Freedom4

Denominator Degrees of Freedom9

P-Value:0.050

F Critical Value:3.633

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F Data

0.0474997890

0.36851522060.0343745398

0.5621432266

0.6501395955

0.6750848247

0.663290885

0.6310389919

0.5883712874

0.5414240785

0.4938788686

0.4478728006

0.4045740045

0.3645471127

0.3279862159

0.294863456

0.2650236028

0.2382438923

0.2142714629

0.1928463347

0.1737150746

0.156638489

0.1413955232

0.1277847843

0.1156246131

0.1047523037

0.0950228563

0.0863075089

0.0784922007

0.0714760606

0.0651699734

0.0594952532

0.0543824345

0.0497701842

0.0456043305

0.0418370006

0.0384258593

0.0353334387

0.0325265485

0.0299757595

0.0276549507

0.0255409127

0.0236130005

0.0218528296

0.0202440099

0.0187719123

0.0174234649

0.0161869734

0.0150519637

0.0140090444

0.0130497846

0.0121666075

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Smooth1 Data

0.010.04749978943.63309027310

0.100.368515220693.63309027310.0343745398

0.200.5621432266

0.300.6501395955

0.400.6750848247

0.500.663290885

0.600.6310389919

0.700.5883712874

0.800.5414240785

0.900.4938788686

1.000.4478728006

1.100.4045740045

1.200.3645471127

1.300.3279862159

1.400.294863456

1.500.2650236028

1.600.2382438923

1.700.2142714629

1.800.1928463347

1.900.1737150746

2.000.156638489

2.100.1413955232

2.200.1277847843

2.300.1156246131

2.400.1047523037

2.500.0950228563

2.600.0863075089

2.700.0784922007

2.800.0714760606

2.900.0651699734

3.000.0594952532

3.100.0543824345

3.200.0497701842

3.300.0456043305

3.400.0418370006

3.500.0384258593

3.600.0353334387

3.700.0325265485

3.800.0299757595

3.900.0276549507

4.000.0255409127

4.100.0236130005

4.200.0218528296

4.300.0202440099

4.400.0187719123

4.500.0174234649

4.600.0161869734

4.700.0150519637

4.800.0140090444

4.900.0130497846

5.000.0121666075


Help

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Page &P

Smooth1

ObsYearMonthPercent DiffForecastThetaTheta NormDiff Squared0.18

11981Feb2.8832.8830.00000000010.0000000015025.2234841628

21981Mar3.0052.883021736100.00000000030.0149757534

31981Apr-0.6102.905049319100.000000000412.3530458635

41981May-0.6012.272405117300.00000000058.2584859718

51981Jun-1.4991.755128648200.000000000610.5917601551

61981Jul-2.5121.169318927400.000000000713.5526927083

71981Aug-7.4420.506667251400.000000000963.1760649312

81981Sep-3.572-0.924033452400.00000000117.0140651324

91981Oct0.302-1.400746899100.00000000142.900573299

101981Nov4.273-1.09418764730.00000000010.000000001728.8073812655

111981Dec-1.573-0.12808250140.00000000010.0000000022.0865984747

121981Jan-0.446-0.38809363690.00000000010.00000000250.0033201392

131982Feb-5.362-0.39846535360.00000000010.00000000324.6385200954

141982Mar-0.197-1.29193502360.00000000010.00000000371.199958318

151982Apr3.110-1.09475832740.00000000020.000000004517.6818865907

161982May-3.397-0.33786128980.00000000020.00000000549.3592028839

171982Jun-0.929-0.88853194770.00000000020.00000000660.0016030204

181982Jul-0.410-0.89573874050.00000000030.00000000810.235812082

191982Aug11.465-0.80832986530.00000000040.0000000099150.6467993811

201982Sep-0.5611.40095877580.00000000040.0000000123.8508718475

211982Oct10.6521.04773328430.00000000050.000000014792.2450468292

221982Nov4.7962.77653041450.00000000060.00000001794.0770800042

231982Dec0.6993.13998246380.00000000080.00000002185.9605410005

241983Jan2.7862.70052651780.0000000010.00000002660.0073612868

251983Feb3.4322.71597014740.00000000120.00000003250.5129618449

261983Mar1.5652.84488858580.00000000140.00000003961.6386907017

271983Apr8.5102.61446814680.00000000170.000000048334.7619361174

281983May-2.1383.67573471560.00000000210.000000058933.8031565237

291983Jun1.8322.62920603880.00000000260.00000007180.6360203561

301983Jul-1.8612.48565444690.00000000320.000000087518.8929242464

311983Aug1.4131.70326659840.00000000380.00000010680.0844958807

321983Sep1.3951.65094388310.00000000470.00000013020.065315143

331983Oct-0.6431.60494159370.00000000570.00000015885.053596563

341983Nov4.1481.20029788790.0000000070.00000019368.6883241452

351983Dec-1.3621.73086522790.00000000850.00000023619.5661098649

361984Jan-3.0241.17414091580.00000001040.00000028817.6235375516

371984Feb-5.4030.41849376490.00000001260.000000351233.8917571767

381984Mar0.889-0.62940553010.00000001540.00000042832.3043296985

391984Apr0.503-0.35616519220.00000001880.00000052230.7382538231

401984May-5.629-0.20150613540.00000002290.000000636929.456282668

411984Jun2.494-1.17843169980.0000000280.000000776713.4834581286

421984Jul-1.512-0.51747478490.00000003410.00000094720.9887488057

431984Aug9.782-0.6964593130.00000004160.0000011551109.8043301441

441984Sep-1.4431.18971679130.00000005070.00000140876.9321423811

451984Oct0.0560.7157954760.00000006180.00000171790.4359599169

461984Nov-1.5270.59694640650.00000007540.00000209514.5123120466

471984Dec1.9030.21458674560.0000000920.00000255492.8520095394

481985Jan6.2070.51856883240.00000011220.000003115832.3562303852

491985Feb-0.2141.5424545020.00000013680.00000379973.086855902

501985Mar-1.3421.22620439330.00000016680.00000463386.5951080975

511985Apr-0.6880.76394742760.00000020340.0000056512.1091953033

521985May4.5590.50253218740.00000024810.000006891516.451735648

531985Jun1.5241.23262549310.00000030260.00000840420.0850387819

541985Jul0.8981.28511603060.0000003690.00001024910.1499997798

551985Aug-0.9971.21540238160.000000450.00001249894.8966696715

561985Sep-0.4030.81709082340.00000054870.00001524251.4893423577

571985Oct3.4380.59742131620.00000066920.00001858848.0696144433

581985Nov7.1181.10874851870.00000081610.000022668836.1081425043

591985Dec5.0632.19036943990.00000099520.00002764498.2543707905

601986Jan1.5722.70751701420.00000121370.00003371331.2884670158

611986Feb8.7892.50319782440.00000148010.000041113839.507859175

621986Mar6.4103.63459282050.0000018050.00005013877.7026427222

631986Apr-1.9044.13415825590.00000220120.000061144836.4617892725

641986May5.1983.04725348890.00000268440.00007456684.6254018328

651986Jun0.8533.43437494190.00000327370.00009093526.6630389219

661986Jul-6.2032.96974340940.00000399230.000110896584.1436593988

671986Aug6.9301.31860605450.00000486860.000135239731.4883849206

681986Sep-6.8882.32866727620.00000593740.000164926584.9492286672

691986Oct6.2360.66964497110.00000724070.000201129830.9866457693

701986Nov1.9391.6716266690.00000883010.00024528030.0717525567

711986Dec-0.9551.71984267080.00001076840.00029912237.1545327716

721987Jan13.8241.23837942010.00001313220.0003647833158.3897097563

731987Feb3.0563.50373294570.00001601490.00044485770.2004524421

741987Mar3.6293.42314349770.00001953030.00054250930.0422178964

751987Apr-0.7953.4601280870.00002381750.000661596718.1089637909

761987May0.2282.69414477980.00002904570.00080682536.0824958714

771987Jun5.5402.25021588210.00003542160.000983933310.8246957966

781987Jul6.3482.84243218030.00004319710.001199918712.2924022371

791987Aug3.5333.47352158160.00005267940.00146331540.0035783449

801987Sep-2.5043.48428904990.00006424310.00178453135.8549878764

811987Oct-23.2162.40646642670.00007834530.0021762573656.5061181925

821987Nov-8.025-2.2055611350.0000955430.002653972433.8654118563

831987Dec5.742-3.25305306530.00011651590.003236551780.9085846479

841988Jan1.000-1.63396747690.00014209250.00394701426.9382465789

851988Feb5.791-1.15983754840.00017328360.00481343248.3137733271

861988Mar-4.0340.09130842870.00021132140.00587003917.0145253482

871988Apr2.227-0.65116757860.0002577090.00715858418.282662617

881988May-0.060-0.13313450160.00031427930.00872998060.0054165076

891988Jun5.445-0.11988705480.00038326740.010646317930.9655109185

901988Jul-0.6060.88175287730.00046739930.01298331452.213580541

911988Aug-4.5600.61394695970.00056999920.015833310326.7745456457

921988Sep4.000-0.31744730640.00069512090.01930891518.6378012179

931988Oct1.6920.459640050.00084770850.02354745731.5174937859

941988Nov-1.5890.68137592590.00103379080.02871641145.1541739435

951988Dec2.5570.27272542020.00126072050.03502001395.2161775704

961989Jan8.0120.68382656750.0015374640.04270733453.7049452825

971989Feb-3.5832.00293241130.00187495610.052082114631.2048704324

981989Mar0.8460.9974286560.00228653190.0635147740.0230108936

991989Apr6.2050.97012384040.00278845350.077457041427.4001338967

1001989May2.5361.91233631020.0034005530.09445980660.3894326197

1011989Jun-1.6162.02466447620.00414701590.115194886113.2576016806

1021989Jul9.0411.36926666230.00505733650.140481568458.8518298649

1031989Aug2.8792.75013569580.00616748350.17131898590.0166991172

1041989Sep-1.6242.77339620740.00752132130.208925592519.3360444453

1051989Oct-1.7731.9818863590.00917234310.254787307914.0981401175

1061989Nov2.3131.30603153750.01118578430.31071622921.0146936702

1071989Dec1.7341.48734914530.01364120030.37892223070.0608964142

1081990Jan-5.9081.53176810320.01663561010.462100281455.3506596107

1091990Feb1.4170.19260367290.02028732940.56353692851.4993399849

1101990Mar3.0430.41300924340.02474064560.68724015676.9194107835

1111990Apr-1.8640.8864951560.03017151910.8380977527.5627052981

1121990May8.2770.39148843710.03679453551.022070429362.1812776487

1131990Jun0.1401.81088034770.04487138471.24642735282.7915302793

1141990Jul0.8511.51013862960.05472120091.52003335710.4346775573

1151990Aug-10.0111.39146448390.06673317181.853699216130.0165322727

1161990Sep-6.192-0.6609817750.08138191682.260608830.5916670417

1171990Oct-0.414-1.65655699180.099246242.756841.544278963

1181990Nov4.804-1.43287275360.1210323.36238.8937129926

1191990Dec2.891-0.31030591770.18510.2509908057

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Smooth2

Exponential Smoothing (1 Parameter)

Smoothing Factor0.180Mean Square Error25.223

Final Value-0.310Standard Error5.022

&A

Page &P

Smooth2

2.88302173612.883021736100.0000000015

3.00539719672.883021736100.0000000003

-0.60964069052.905049319100.0000000004

-0.60135304432.272405117300.0000000005

-1.49936980091.755128648200.0000000006

-2.51207927281.169318927400.0000000007

-7.44166999180.506667251400.0000000009

-3.5724414898-0.924033452400.0000000011

0.3023600555-1.400746899100.0000000014

4.2730631635-1.094187647300.0000000017

-1.5725888096-0.128082501400.000000002

-0.4457142857-0.388093636900.0000000025

-5.3621857422-0.398465353600.000000003

-0.1965089339-1.291935023600.0000000037

3.1102252148-1.094758327400.0000000045

-3.3971427224-0.337861289800.0000000054

-0.9285696854-0.888531947700.0000000066

-0.4101338785-0.895738740500.0000000081

11.4654959189-0.808329865300.0000000099

-0.5614050661.400958775800.000000012

10.65216178521.047733284300.0000000147

4.79570846612.776530414500.0000000179

0.69856054193.139982463800.0000000218

2.78632445962.700526517800.0000000266

3.43218369432.715970147400.0000000325

1.56477503552.844888585800.0000000396

8.51039352942.614468146800.0000000483

-2.13831348883.675734715600.0000000589

1.8316971952.629206038800.0000000718

-1.86094471182.485654446900.0000000875

1.41258484681.703266598400.0000001068

1.39537560851.650943883100.0000001302

-0.64307899411.604941593700.0000001588

4.14789422141.200297887900.0000001936

-1.36204761681.730865227900.0000002361

-3.02389881141.174140915800.000000288

-5.40316898520.418493764900.0000003512

0.8885963469-0.629405530100.0000004283

0.5030517903-0.356165192200.0000005223

-5.6288703822-0.201506135400.0000006369

2.4935511608-1.178431699800.0000007767

-1.5118332745-0.517474784900.0000009472

9.7822968223-0.69645931300.0000011551

-1.44317940511.189716791300.0000014087

0.0555228680.71579547600.0000017179

-1.5272739320.596946406500.0000020951

1.90337611650.214586745600.0000025549

6.20682255260.518568832400.0000031158

-0.21449054611.54245450200.0000037997

-1.34188986071.226204393300.0000046338

-0.68835946260.763947427600.000005651

4.55860610780.502532187400.0000068915

1.52423959071.232625493100.0000084042

0.89781798031.285116030600.0000102491

-0.99743960811.215402381600.0000124989

-0.40329532760.817090823400.0000152425

3.43812799650.597421316200.0000185884

7.11775363641.108748518700.0000226688

5.06341151942.190369439900.0000276449

1.57241040432.707517014200.0000337133

8.78872558072.503197824400.0000411138

6.40995635033.634592820500.0000501387

-1.90420156054.134158255900.0000611448

5.19792822793.047253488900.0000745668

0.85308864983.434374941900.0000909352

-6.20324189532.969743409400.0001108965

6.93005728581.318606054500.0001352397

-6.88812330772.328667276200.0001649265

6.23620995940.669644971100.0002011298

1.9394933461.67162666900.0002452803

-0.95495316651.719842670800.0002991223

13.82367678471.238379420100.0003647833

3.05601379033.503732945700.0004448577

3.6286134383.423143497700.0005425093

-0.79533473053.46012808700.0006615967

0.22787312582.694144779800.0008068253

5.54030642752.250215882100.0009839333

6.348484412.842432180300.0011999187

3.533340853.473521581600.0014633154

-2.50361441263.484289049900.001784531

-23.21590891582.406466426700.0021762573

-8.024960748-2.20556113500.0026539724

5.7418668703-3.253053065300.0032365517

1.0000876817-1.633967476900.0039470142

5.7909734351-1.159837548400.004813432

-4.03355827810.091308428700.005870039

2.2267939599-0.651167578600.0071585841

-0.0595375751-0.133134501600.0087299806

5.4447792351-0.119887054800.0106463179

-0.60605777630.881752877300.0129833145

-4.56046562970.613946959700.0158333103

3.9997046735-0.317447306400.019308915

1.69150602720.4596400500.0235474573

-1.58890466110.681375925900.0287164114

2.55662068280.272725420200.0350200139

8.01219236640.683826567500.042707334

-3.58319956282.002932411300.0520821146

0.84573523620.99742865600.063514774

6.20463756150.970123840400.0774570414

2.53638167691.912336310200.0944598066

-1.61643448982.024664476200.1151948861

9.04076129281.369266662300.1404815684

2.87936076012.750135695800.1713189859

-1.62388072792.773396207400.2089255925

-1.77286264961.98188635900.2547873079

2.3133515811.306031537500.3107162292

1.73412113351.487349145300.3789222307

-5.90803428741.531768103200.4621002814

1.41707906460.192603672900.5635369285

3.04348653540.413009243400.6872401567

-1.86354217070.88649515600.838097752

8.27699905150.391488437101.0220704293

0.14009302451.810880347701.2464273528

0.85083782011.510138629601.5200333571

-10.01101473221.391464483901.853699216

-6.1919552013-0.66098177502.2606088

-0.4138667797-1.656556991802.75684

4.8036096678-1.432872753603.362

2.8914109351-0.310305917705

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Weights for Smoothing Factor

Observed and Forecasted

Percent Diff

Forecast

Month

Theta Norm

Smooth2 Data

Exponential Smoothing (2 Parameters)

Smoothing FactorsMean Square Error31,035.85

Location0.15Standard Error176.17

Trend0.40Final Observation2,633.66

Forecasted Trend-7.36


Help

&A

Page &P

Smooth2 Data

636.46146831960947.2700

698.36568396090974.5800

750.606192900401003.870

796.83818088550997.750

831.55320390470991.750

857.96967341030976.880

876.34619245350952.340

886.59684561830881.470

880.88656799530849.980

871.39643232780852.550

865.81117719770888.980

869.817326449508750

870.0744229810871.10

869.68207683320824.390

Documents

Estimation Chapter 7