You are given a list of 2000 measurements (listed under the question) You are to
ID: 3229641 • Letter: Y
Question
You are given a list of 2000 measurements (listed under the question) You are to construct a test of goodness of fit of your measurements with the distribution (PDF)
P(x) = 1/1.45 EXP(x/1.45)
(A 2 test is one of many possibilities.) You must compute the test statistic and its p-value for your measurements. Are your measurements consistent with the “theory”?
2.641975065
1.42431858
0.630375486
0.430782199
0.592991677
7.785120778
0.322007315
1.693892126
0.953928548
1.023372994
1.664433691
3.226439226
0.541227763
0.72062351
0.952614793
0.659690177
0.502965549
2.808031194
0.458112617
3.058588242
0.822942459
7.656021005
0.506123486
0.519048561
5.863739576
1.238058759
0.952501896
1.446457505
0.629774163
2.176000744
1.219015594
0.260678061
1.803608513
1.207371761
0.573684286
0.370317836
1.119351743
5.608194201
0.371926555
0.118912127
1.715537679
3.674301616
2.91005038
3.081402515
0.364127178
0.205600467
4.780372582
7.275656115
0.399516425
1.741812262
0.897143931
0.120721297
9.701980813
0.429809273
0.11746403
0.113008365
0.56280719
0.201909148
0.742282221
2.705883381
0.963233858
0.094270919
0.662207133
0.004430556
0.028000398
0.184274503
0.043090847
9.161194777
0.112880586
0.116997188
0.532386185
0.50642693
0.798028172
1.654554441
0.927536116
12.84546845
0.05888872
2.892414344
0.846412638
0.829315259
0.072485097
0.150546887
0.546752796
0.204176056
0.749499497
4.270852785
2.096155351
1.788274405
1.893556819
6.620881697
1.816963976
1.15277959
0.904521575
0.551987325
0.211702269
5.676013412
0.074522898
1.543607082
8.143183309
0.085759603
0.513928514
1.086310368
1.466221671
1.576080967
0.075199177
0.735691837
1.432904246
2.944148213
0.218807777
0.616210827
0.867032083
3.084285565
5.695314935
0.341674496
0.488809013
0.786135837
0.682237091
1.736254986
1.052606713
12.43441491
1.16599571
0.491914575
2.116567886
0.432057633
0.369403131
2.51205268
6.332072008
0.666244125
1.351539556
0.800143517
0.731843115
1.86762562
1.972416027
0.290535879
0.11514503
2.306363676
1.039390728
0.85090402
7.742738879
1.075538827
1.272553304
8.011076538
2.628893939
0.901574061
1.62727799
0.807917108
1.912457103
1.330582396
4.809039702
0.360157766
0.172385737
1.565900873
6.718255762
1.573196611
6.432521026
2.855250378
0.967394184
0.313577795
0.623211158
0.627869752
0.268083313
0.002388437
1.190896945
0.92057818
0.023623369
0.829155881
0.614746639
0.922848749
0.19146422
0.55169845
2.316675775
1.810995665
0.452904937
0.031368424
0.19594159
1.065427448
3.789025291
3.198579015
0.512255883
0.941201821
1.207657528
0.726498711
0.94919261
0.18211328
7.308856394
0.522348095
1.271284154
10.088052
0.152345737
2.221810483
0.408668431
4.790535182
0.324915494
3.011678352
2.517752327
1.428664754
2.086112945
0.21733977
0.562870571
0.721656459
0.996135087
0.189885667
2.182927249
0.398413782
0.380045415
5.262452801
0.369097166
2.927516217
0.045339445
1.419317421
1.280511771
3.157200482
0.283985787
0.13190431
8.808580857
0.168129685
4.280256131
0.893695413
5.146926822
1.969803301
0.171361035
0.255271274
1.557696926
0.599357701
2.401102741
0.845294039
0.159163431
0.385903365
1.893090707
2.312882813
1.117773899
0.47388353
9.526746791
2.896688264
0.550483612
1.741937407
2.594129839
0.345030504
15.72071979
2.842534206
0.40401899
1.924748214
0.440235621
0.080788798
0.23791029
2.785987106
0.854901771
0.193642948
1.046292255
2.862953128
0.788376438
1.990397692
10.85676024
7.247339819
1.351335663
0.146021305
1.252777498
0.84997612
1.159941943
2.67503272
0.784665514
7.497520826
0.280538115
0.220790037
2.107643666
1.069597609
0.59092703
0.433112176
2.292036325
0.438591925
0.294034483
0.560719028
0.021813602
1.104649225
1.947104768
0.537694275
0.232360276
0.155479092
2.013199431
0.563792006
0.328240365
1.669004827
1.769915874
1.195305794
0.816847295
3.018453308
2.492749797
1.632400664
3.820264381
1.334559488
2.00076218
2.263694408
3.192958276
0.514203764
0.81885762
3.209715062
5.153550814
2.895814171
0.988485687
1.494893237
2.444412638
1.420922577
0.427720164
1.369733923
0.609958176
0.922353641
0.173736305
0.556545702
0.372786939
0.840924558
0.077556534
0.17016873
1.744348647
1.568809854
0.978756707
3.523779284
1.318974208
0.285779215
0.865875325
1.734002715
0.045680103
1.206918682
0.293380495
0.00495124
11.91103182
0.345637829
0.480365675
0.20293261
2.316244303
0.206552954
0.463479131
0.247927342
0.071870451
0.332915917
7.445316606
0.222566475
0.027233434
4.875455232
0.390077878
0.700544221
0.281603275
1.185669561
0.800390687
0.220043127
0.576257886
0.044750546
6.426747274
0.750271206
0.454678327
0.459280876
0.185380293
0.361875438
1.637903645
1.268913404
1.125341789
0.119239595
0.499284215
0.199326646
0.854008511
1.069100834
0.680063986
0.615885894
0.786174287
1.447108847
1.881782694
0.64840182
1.908454756
3.135443083
0.505497256
2.39218414
0.36387024
0.394984396
1.001484526
7.222867193
0.015005792
3.766059149
2.998913077
0.110121067
0.706088715
1.165405523
0.83229383
1.67506102
2.238080751
0.633231162
0.46489953
0.007407619
2.645294804
3.459172008
1.169907184
25.02648162
0.330448735
0.5887869
0.549780107
1.749949125
0.97860617
1.268028612
2.265404977
0.1691571
0.798626549
0.752768368
0.191377103
1.598149235
3.105650158
2.088226767
0.268499679
0.047401806
0.230211964
0.663787485
0.149183697
0.104004609
0.791580327
2.778447166
0.731478093
0.203690267
0.273875967
1.556378281
1.123952402
0.082541471
0.319317831
2.380607317
0.039601273
0.057715373
4.268395247
0.169075078
11.88930259
0.75298172
0.0945724
0.436630956
0.013822262
0.61307844
0.378335367
0.615662151
0.392054449
0.795277104
5.586523734
1.252635457
0.706467849
1.651086607
1.221627293
1.985033761
0.031732343
0.7410183
0.880518531
1.987069635
6.652583576
9.171771003
0.589914827
0.390843241
1.253353021
0.136329958
1.063661002
0.343367272
0.318602635
4.008512307
0.842519491
0.038598578
0.552081037
0.505908591
3.749818062
0.060288738
3.748016389
0.714753318
2.984393765
0.966276447
2.364962606
2.801886902
0.445210631
0.028489866
6.313230688
0.538359671
0.245944062
0.0636834
0.580178469
1.376335048
2.583734423
2.305430068
0.571782154
0.1706557
6.777970974
0.645136118
0.024039046
1.50186821
1.875829964
1.07522854
0.420554789
0.083733068
0.247265716
0.213820015
2.068212637
1.346802886
2.630160586
0.385049783
0.17010297
0.511022965
2.329715502
0.964250312
2.93029968
0.151828393
0.251624489
0.062559672
2.463557857
1.568295821
0.325743312
0.214979793
1.833193862
2.113340335
0.234901345
0.171351208
1.279229764
0.202302041
2.172890317
0.554645448
0.862036815
1.495785499
1.501215506
4.823564421
0.576843572
1.249741198
0.364365993
0.834325093
0.77109374
1.659916854
11.38891936
1.636856508
0.663781871
0.337451595
0.328878607
1.928664025
0.30042684
1.272453865
2.048287364
0.184880347
0.968500623
0.13487563
2.122309054
1.550064081
2.808810536
1.027945219
4.540628349
1.130634424
1.336195271
3.304938649
3.4563897
0.438131707
0.896309179
0.026910061
0.792344481
2.456648776
2.618104273
0.922712874
4.798212643
1.525167398
3.141302222
2.390851882
0.625612835
1.008967791
0.692352122
15.18279535
0.296415311
0.020213721
0.248756069
1.151663376
0.247257091
0.678495251
2.199022716
1.024411983
1.695651952
2.596351866
0.015115846
0.387484035
0.506807057
1.629680856
3.338100365
0.573412671
1.48632799
0.30934837
1.296849192
1.353857863
0.608043364
0.340639136
5.473599809
3.150170608
1.278946103
4.940186439
1.107338457
0.319169311
2.945951101
1.870028229
4.014765347
1.662457164
0.583259733
0.166072155
0.14848306
1.621607612
1.662011513
0.353431398
2.185456007
1.563100233
0.257772854
2.16565943
0.353452018
3.947514458
0.433436079
4.121901262
1.270798331
0.234847153
0.671911772
4.018047208
1.110393493
1.016968204
0.522791385
0.126901249
1.107186265
1.918378349
0.897538726
0.650086953
2.174657391
0.459604419
0.574719289
0.179283381
0.565587194
1.338039163
1.333827268
7.661672566
2.544404175
5.250950739
2.47222133
0.35746291
0.610737892
0.749606245
2.779653355
4.644091558
0.06979902
1.076723861
4.353759005
1.282974017
0.386446413
0.671357649
1.061537692
1.455282984
0.448517248
0.004190193
3.661070518
0.98738907
0.264652057
0.109050422
3.41807646
1.022764357
1.844917997
1.190105414
2.231082458
0.945691074
4.946468413
0.056936943
0.676315067
1.35747889
2.017820967
0.26606275
2.069206095
22.7865127
2.583327126
0.286942893
0.637569101
3.06716425
0.249382213
5.680851088
0.864888686
1.099676926
1.608445659
2.355348177
0.971246341
0.705683971
0.866215431
1.329213371
0.722836025
4.745998382
1.214107438
0.281970186
1.865415202
0.060226236
0.556217105
4.105727869
0.117591673
2.308270686
5.371592292
0.28773708
0.232286268
0.070588511
1.583431951
1.554261433
0.359052357
0.793306171
0.76003852
1.187346902
0.951166331
6.706842222
0.191293681
5.58190118
0.751345693
0.862258392
1.232555137
0.494649675
0.7011745
0.380421961
3.405102621
0.803126045
0.195073164
1.508408176
9.815514144
0.01858053
5.856530573
0.428622551
0.585642249
1.238716326
1.384281362
0.260828016
6.263368321
0.631925636
1.100113278
1.021546809
0.089792961
2.17582848
1.035054736
0.106259119
0.388982715
0.20171917
0.742463288
1.192304057
3.158580544
0.547042369
0.259856127
2.105428354
1.240835108
0.006415746
2.028839583
3.143170739
1.4934814
0.868725236
0.141197951
0.104937089
0.14478349
2.151057561
2.6409665
2.352737227
2.215560956
12.02276658
3.344079107
0.332780028
0.185682463
3.847837183
1.800603871
0.520175198
0.539137517
1.18401786
6.814794816
0.22122089
1.165867572
1.633146919
0.244026424
1.28873358
1.913235534
4.804490998
1.239846469
1.763062231
0.405733822
0.316832807
0.352184547
0.92977722
0.213964377
0.134127897
0.656354976
0.306411035
0.130070674
4.351382623
0.225086501
0.759054267
0.232910692
3.374719595
0.138816671
1.027518826
2.903397976
0.81132394
1.517447529
4.855041228
0.66225895
9.410390823
0.18429497
0.509967097
1.467866683
0.815448526
0.216934771
1.539859144
0.732896042
8.493576663
0.950190871
0.018744515
2.43871381
0.417245252
0.134289798
0.205167113
1.350702754
0.168946787
2.683648579
0.047590649
1.155881002
1.651404408
0.050099296
0.669085593
4.57537539
0.076755975
0.35820317
1.476742907
0.799030603
0.146351215
3.07152938
0.283943466
1.230805591
1.490087262
14.56062149
0.348123853
2.154774384
0.978148742
2.85274107
0.268933174
1.532098163
2.274761098
1.450867455
1.241471205
0.391229245
0.806221115
0.64944145
4.711027377
0.989043273
3.323098247
0.913465359
1.959980891
0.685069582
1.777378358
1.015445139
0.153890578
3.13668894
0.623029961
0.129853041
2.592482321
2.859883924
2.814570999
2.7776352
1.452621245
1.635262391
0.175678556
1.427898041
0.922698303
0.817677175
5.996703997
1.175034987
1.111688256
0.283601065
0.107379576
0.142264173
1.330773244
1.64802025
0.458118327
1.103334197
0.913292086
2.173800225
0.511221808
0.982357036
0.689975726
2.949528125
0.62048057
1.821335834
2.719292576
2.018907747
0.317334758
1.07731058
0.034226073
0.370108325
2.929107172
1.255128563
1.006053104
0.729571191
2.731773858
0.436183281
3.150349865
1.581477102
0.058075516
1.664454311
0.103070323
0.005336857
0.486692044
0.190378013
0.004262016
1.429236571
0.66544874
0.016914296
0.163785003
1.814318265
0.046140203
2.306830672
3.392311075
4.141886553
0.185907227
1.262162805
5.698599029
7.077754988
5.073017189
0.066275037
1.53321442
1.044371222
0.182156335
0.83710043
0.927155421
0.855979455
0.161894837
0.181593283
2.64938695
0.797377968
1.529329473
0.74222982
0.218365021
0.767434882
0.854883084
0.118059685
0.940020896
0.839522867
0.907430187
1.067999781
3.278634001
3.106610881
1.98201118
0.696672192
0.317953329
0.977976175
0.416801338
0.196374464
7.404832696
0.512013876
0.640572178
1.601545654
1.115561503
3.998941102
1.230515691
0.576611278
0.383984277
7.47507391
3.051615939
2.124631348
1.250800371
0.512781796
5.464587952
3.053216712
0.09346728
0.75736864
0.76981069
1.146445184
0.765157617
1.075862262
0.923127432
0.023265016
2.022238222
0.378823339
0.090106664
2.773122641
1.419528445
0.312483727
0.122441763
0.431467824
0.067114469
5.508893842
3.248603873
0.137968188
1.24990495
7.331790922
0.88950299
4.368852588
1.172056073
0.242357579
0.237240442
4.265767335
2.13577005
4.836797248
0.611922513
1.663191052
0.286301615
0.096016244
4.489508718
0.436575664
1.510522014
2.273932101
0.16999013
0.602242566
0.238740303
1.148322395
7.581918489
0.434894861
0.040171693
1.007368107
0.16005558
0.400476156
0.032097194
2.890732683
2.007033644
3.098362289
0.748839469
1.838111699
1.26503356
0.121650841
5.599955514
0.459981022
0.101419928
0.692604009
0.064094399
0.161177967
0.242155393
1.209422807
2.066150853
3.878759724
0.549950287
1.018562529
0.492328183
0.245802798
0.567051025
0.157177662
0.619330411
1.84297989
0.71912877
0.239190119
2.047610557
0.837588755
0.978267778
4.58100968
0.102584883
0.905870682
0.262545571
0.26276282
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8.369678672
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10.64545874
0.44199018
1.981520181
1.210356891
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3.52110458
8.901713766
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4.180016357
1.379904046
1.668593803
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2.680136846
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12.629838
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1.869340542
0.162176147
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4.910001899
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0.271518441
0.082819969
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1.893397327
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1.198887668
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0.028736073
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1.184275566
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1.875928341
0.463711773
4.878670586
0.251194184
10.11989255
1.589254147
3.082089103
2.349853621
0.477796612
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0.896934088
2.026006111
0.06030864
1.842049602
4.028236807
0.620964628
0.181840866
1.709963136
1.24557359
0.662826644
0.365798781
2.648357642
0.41629788
1.383504344
0.527178877
0.515961496
0.411134145
1.053180449
0.883473385
0.083459029
1.066451713
1.39224259
0.934883452
0.682876445
0.101318775
0.02391955
0.293060419
1.378232772
1.304327134
4.488930828
1.018918339
1.029810168
1.036071989
0.07748193
0.355016779
0.086743923
0.530384075
0.464881774
Explanation / Answer
The percentile of the given measurements are first obtained:
> measurements <- read.csv("clipboard",header=F)
> dim(measurements)
[1] 2000 1
> quantile(measurements$V1,probs=seq(0,1,.1))
0% 10% 20% 30% 40% 50% 60% 70% 80% 90% 100%
0.000542265 0.143866573 0.296270762 0.478471071 0.693267991 0.932099186 1.252472638 1.704385588 2.386502696 3.869368716 25.026481620
Now, if the distribution is f(x) = 1/1.45 exp(x/1.45), that is exponential distribution, the expected frequencies within the empirical intervals is
> diff(2000*pexp(emp_quantile,1/1.45))
10% 20% 30% 40% 50% 60% 70% 80% 90% 100%
188.16222 180.69464 192.51917 197.97542 188.29302 208.47176 225.76867 231.67551 246.98359 138.70812
> exp_freq <- diff(2000*pexp(emp_quantile,1/1.45))
> obs_freq <- rep(200,10)
> sum((obs_freq-exp_freq)^2/exp_freq)
[1] 47.483947
Since the chi-square value is large enough, we don't have enough evidence to reject th null hypothesis and hence by the goodness-of-fit test, we conclude that the sample is from expnential distribution with rate 1/1.45.
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