Simulation in R Software (Statistics and R Programming) Suppose that three syste
ID: 3125108 • Letter: S
Question
Simulation in R Software (Statistics and R Programming)
Suppose that three systems (A,B, and C) are independently operating each under the heading of an engineer. From each system there are 5 products randomly sampled for quality control. Assume that the quality is normally distributed and all manufactured products are independent.
a. Suppose that each system has quality mean 8.4 with standard deviation 0.05. Simulate this situation one thousand times and test each H0 : µ1 = µ2 = µ3 by estimating the linear model and its F-test, taking significance level = 0.05. Note that you do not have to construct the matrix X; R can easily do the work. Record the probability for a false positive (incorrect decision for differences in mean). Hint: Use R functions such as rnorm, lm, and anova.
b. Suppose that system A and B have quality mean 8.4, but C has mean 8.45. Take everything else as before. Use 1000 random samples to record the probability for a true positive.
c. The foregoing situation seems not completely satisfactory. How would you increase the probability for true positives?
Explanation / Answer
a.
P=c()
for ( i in 1:1000){
A=rnorm(5, 8.4,0.05)
B=rnorm(5, 8.4,0.05)
C=rnorm(5, 8.4,0.05)
Y=c(A,B,C)
X=gl(3,5)
M=model.matrix(Y ~ X)
summary( lm(Y~X))
P[i]=anova(lm(Y~X))$Pr[1]
}
PF=P[P>=0.05]
[1] 0.62650548 0.17281606 0.94108930 0.89522135 0.37784552 0.18002791
[7] 0.65283814 0.28075983 0.08362619 0.07235984 0.13933871 0.75497506
[13] 0.97330876 0.33616279 0.21573719 0.52394285 0.35832521 0.54515014
[19] 0.44088159 0.11940592 0.97800493 0.86742019 0.76480411 0.70157192
[25] 0.80066658 0.97876751 0.65419271 0.67363032 0.45512517 0.98688841
[31] 0.65510694 0.65267831 0.23831470 0.55435306 0.81750712 0.76389328
[37] 0.33341275 0.20850609 0.91439625 0.39443945 0.78848143 0.77743454
[43] 0.67591987 0.78671619 0.95436733 0.53611879 0.34762622 0.08673239
[49] 0.51613137 0.68183835 0.38397531 0.79368682 0.97934141 0.74058513
[55] 0.28077443 0.50797529 0.35150443 0.20660428 0.51973753 0.57532269
[61] 0.84637098 0.90518013 0.57190382 0.19864343 0.31900463 0.07626833
[67] 0.94968840 0.12805937 0.84515985 0.46603226 0.10974079 0.15290605
[73] 0.07481965 0.65692349 0.79515072 0.49524236 0.61805852 0.12704018
[79] 0.51817732 0.11023255 0.33362495 0.65943412 0.91443749 0.78218348
[85] 0.87785199 0.47192280 0.59483604 0.96409964 0.65079808 0.91245173
[91] 0.30265522 0.59718805 0.40539350 0.38717787 0.37764479 0.33871800
[97] 0.51935808 0.07388590 0.61986893 0.24243488 0.62389140 0.28832123
[103] 0.48950727 0.63141792 0.49687909 0.67046699 0.34730897 0.40978701
[109] 0.29999239 0.94795539 0.68890834 0.41596551 0.75284234 0.25731816
[115] 0.94780738 0.60809769 0.12205779 0.44584297 0.27616649 0.70264771
[121] 0.80502290 0.90288097 0.15821086 0.16731261 0.91086667 0.17808924
[127] 0.18897023 0.48155989 0.53373435 0.10124007 0.44453509 0.36948240
[133] 0.55496819 0.40519893 0.39262978 0.46016993 0.89583168 0.36339207
[139] 0.05033480 0.90374201 0.31086880 0.12788792 0.12691722 0.21079871
[145] 0.27452702 0.57288396 0.16366105 0.18410935 0.83284286 0.50421828
[151] 0.61311390 0.14417486 0.38490819 0.91054895 0.11477014 0.12851453
[157] 0.21019266 0.07877120 0.62804801 0.78209095 0.94453608 0.52942152
[163] 0.92159499 0.97884702 0.88832338 0.22946348 0.26722170 0.39908740
[169] 0.93657941 0.87423344 0.74166847 0.93318909 0.51355238 0.40078696
[175] 0.28033747 0.93793049 0.18764287 0.42084723 0.38232397 0.59483251
[181] 0.75567047 0.44456283 0.52751231 0.60950499 0.55259517 0.84300012
[187] 0.16609218 0.26209166 0.80257145 0.16857709 0.98224695 0.39575089
[193] 0.11080269 0.37862287 0.79911128 0.91874410 0.09448476 0.87766860
[199] 0.09941319 0.28183369 0.89700241 0.36606726 0.50829912 0.97151953
[205] 0.65585161 0.81591637 0.49871725 0.78205906 0.22890649 0.60811511
[211] 0.37192389 0.40103732 0.53035679 0.73337090 0.72368194 0.87651975
[217] 0.15472069 0.13516338 0.36166701 0.45828346 0.05165937 0.90698898
[223] 0.44749047 0.99270820 0.84682002 0.52723709 0.82226272 0.98638058
[229] 0.22486880 0.96305172 0.60678473 0.80953269 0.06650479 0.59937887
[235] 0.38732137 0.38477740 0.34119226 0.95071324 0.48622347 0.27271795
[241] 0.42089164 0.97656085 0.44845721 0.14715922 0.38795361 0.15708706
[247] 0.14497321 0.85154403 0.43970453 0.08618087 0.27708490 0.35686647
[253] 0.75866455 0.16337231 0.63157865 0.42436766 0.26098118 0.94230037
[259] 0.24973174 0.26581511 0.15387657 0.30026093 0.86866784 0.18604044
[265] 0.94583054 0.61766906 0.37203080 0.37660776 0.27113323 0.59620805
[271] 0.83120329 0.58781077 0.60300810 0.24390798 0.09354237 0.67861880
[277] 0.71609765 0.77883204 0.69200149 0.55173166 0.12383949 0.79688563
[283] 0.84116583 0.54205846 0.07271871 0.79883125 0.54974891 0.66450655
[289] 0.66847415 0.65110982 0.07405593 0.24439114 0.57935788 0.65805301
[295] 0.78663974 0.39454305 0.81723306 0.66986996 0.73544547 0.66340651
[301] 0.61959485 0.15045308 0.39387822 0.75998848 0.05754358 0.07994527
[307] 0.38410204 0.94673286 0.63277976 0.53159052 0.45791659 0.72391185
[313] 0.34853565 0.45758922 0.59061792 0.24476412 0.40126438 0.72403098
[319] 0.27438398 0.79133449 0.61198563 0.33173166 0.78320465 0.77528337
[325] 0.67328365 0.74902949 0.42102371 0.15955945 0.56031491 0.87564668
[331] 0.73800339 0.61851078 0.22881080 0.50211651 0.30871419 0.37722544
[337] 0.68406371 0.64031238 0.32662977 0.61047541 0.93987893 0.12060703
[343] 0.48592590 0.71533885 0.47536370 0.38127795 0.52191382 0.31700677
[349] 0.74621303 0.95349094 0.57986987 0.34860968 0.07046579 0.48256119
[355] 0.37895834 0.31857978 0.11600958 0.33227039 0.07493989 0.48533798
[361] 0.19927950 0.24645844 0.73706900 0.80731368 0.76210959 0.79110763
[367] 0.94031990 0.82382796 0.69935528 0.40012450 0.25807174 0.82765355
[373] 0.81227930 0.82130935 0.81911201 0.43194941 0.85132225 0.15938522
[379] 0.47914038 0.65841962 0.45470548 0.48909820 0.64254362 0.12031781
[385] 0.22829863 0.39081235 0.71962795 0.23668370 0.84068277 0.62174723
[391] 0.30216616 0.71947644 0.16355380 0.86011067 0.48063521 0.74340191
[397] 0.67965120 0.98038543 0.05496949 0.46707994 0.22112768 0.69617345
[403] 0.54067990 0.34501283 0.62599358 0.40611820 0.80611337 0.71142132
[409] 0.63924454 0.94801965 0.13291652 0.55119997 0.80141069 0.16531499
[415] 0.96749518 0.74643101 0.97408766 0.56958395 0.11359824 0.52769459
[421] 0.66992501 0.49549504 0.86730108 0.08817762 0.76772553 0.11738458
[427] 0.05923586 0.83572001 0.81504913 0.65896528 0.23065183 0.39020653
[433] 0.60595574 0.27697540 0.30110592 0.43180011 0.98160400 0.18979006
[439] 0.40473819 0.41537172 0.45817585 0.08423146 0.72685956 0.10663413
[445] 0.51132337 0.35728633 0.10461961 0.52189623 0.80161702 0.39947018
[451] 0.31387141 0.40340612 0.34088972 0.37281453 0.97589610 0.77861697
[457] 0.40295040 0.47661052 0.07321099 0.38296656 0.30745699 0.96819759
[463] 0.97803748 0.81870494 0.31980599 0.50777919 0.83917831 0.75738158
[469] 0.84672852 0.84361862 0.91596253 0.90419552 0.24485351 0.58257148
[475] 0.14727215 0.47461788 0.50705615 0.24487488 0.18669769 0.57643382
[481] 0.57829660 0.73336813 0.96110408 0.51716513 0.63714516 0.96001449
[487] 0.19119894 0.81667264 0.12270763 0.40074932 0.08192981 0.84166708
[493] 0.73565519 0.40672421 0.70719623 0.60374982 0.46734018 0.95387605
[499] 0.17288713 0.61337174 0.52477420 0.26661657 0.67424148 0.17122974
[505] 0.48245043 0.35577062 0.47647818 0.82584389 0.32626537 0.45338761
[511] 0.65960840 0.26420722 0.71888791 0.29826015 0.33175032 0.31576712
[517] 0.82050641 0.85764203 0.74802946 0.34752008 0.79335448 0.37128916
[523] 0.63425155 0.29466708 0.55300858 0.76884671 0.40204983 0.32966383
[529] 0.58533312 0.94686167 0.81843720 0.87599580 0.60819353 0.05869085
[535] 0.47402846 0.10255427 0.72790735 0.44747341 0.17954569 0.69787896
[541] 0.36822814 0.38122452 0.41907075 0.24217798 0.24080070 0.21966412
[547] 0.44024629 0.90555093 0.08004844 0.12922961 0.83541711 0.48981065
[553] 0.15114744 0.34081759 0.30848159 0.36837324 0.91435228 0.28229109
[559] 0.20500224 0.65660906 0.16888556 0.10430009 0.85498957 0.82610095
[565] 0.86763591 0.18918940 0.39754071 0.92356015 0.72362237 0.59270412
[571] 0.24588045 0.37325070 0.63749894 0.34514667 0.87076248 0.95272396
[577] 0.54986732 0.26350738 0.66966512 0.31752263 0.83149145 0.47431796
[583] 0.11247503 0.85619121 0.78652294 0.36112682 0.48457173 0.43226180
[589] 0.56157557 0.12608179 0.41366121 0.85608143 0.86873670 0.38464923
[595] 0.16909251 0.48475543 0.64405051 0.79838344 0.64270999 0.87288264
[601] 0.52409609 0.48905840 0.42138327 0.86387013 0.14593472 0.98419550
[607] 0.50196756 0.42018088 0.70357752 0.26533278 0.61691751 0.20849118
[613] 0.14170107 0.98663574 0.30902218 0.89508175 0.27095225 0.17955457
[619] 0.09971955 0.50475227 0.05526979 0.13716868 0.16262081 0.61994061
[625] 0.21603612 0.65289111 0.15362685 0.57967752 0.86322077 0.92286076
[631] 0.25022250 0.40951926 0.74753285 0.92023675 0.36390114 0.53438434
[637] 0.40614934 0.69409987 0.14347529 0.26326388 0.52474883 0.75079864
[643] 0.62807228 0.28011764 0.51972616 0.97877489 0.23615527 0.29490143
[649] 0.23138254 0.29662505 0.11840006 0.90945510 0.43161068 0.50688800
[655] 0.68983943 0.99483071 0.07543272 0.44189677 0.37490283 0.74654494
[661] 0.32903646 0.98283469 0.91066654 0.70336990 0.45926313 0.61179868
[667] 0.24648714 0.96886334 0.88611989 0.95445780 0.22340583 0.62024721
[673] 0.11056476 0.50968959 0.64462575 0.28237969 0.62888039 0.51207902
[679] 0.37936759 0.99605176 0.35503246 0.59339361 0.24689446 0.26117336
[685] 0.24733455 0.90628109 0.11590823 0.14300610 0.41434515 0.23284404
[691] 0.99853914 0.54250222 0.70913050 0.09368022 0.38237153 0.90325138
[697] 0.25355610 0.93636540 0.23774475 0.95618449 0.34599728 0.93107705
[703] 0.55622633 0.90632202 0.61162949 0.99973173 0.18323885 0.29754547
[709] 0.12816862 0.64317477 0.43748221 0.75798158 0.71048311 0.44176171
[715] 0.44527913 0.92975129 0.88920298 0.93769262 0.34753935 0.58718009
[721] 0.08912392 0.73314083 0.09410331 0.12866005 0.74454921 0.35870614
[727] 0.49231336 0.76839307 0.30146743 0.89357948 0.09548198 0.15216777
[733] 0.12871723 0.30893521 0.10186872 0.35016736 0.75459701 0.80038485
[739] 0.78844143 0.41660719 0.19000398 0.06365756 0.13594541 0.46474259
[745] 0.24592106 0.08255072 0.74418999 0.51433960 0.87856552 0.96182944
[751] 0.98391868 0.27407741 0.42585942 0.59517762 0.15588203 0.55795145
[757] 0.91664782 0.44145377 0.78148006 0.05745876 0.79897054 0.10696962
[763] 0.81907964 0.79744653 0.28928341 0.18629339 0.54372115 0.06114164
[769] 0.68175425 0.70509676 0.37219076 0.27919044 0.22629548 0.78479079
[775] 0.79061528 0.39335950 0.71333709 0.46333000 0.27629409 0.12618839
[781] 0.18353833 0.51723595 0.99775847 0.51257874 0.75878417 0.98746977
[787] 0.27281050 0.61009327 0.36234853 0.51780090 0.08184158 0.60243439
[793] 0.48767562 0.55649086 0.39602699 0.20376341 0.94025939 0.07536472
[799] 0.23920209 0.29470099 0.15716612 0.51115856 0.62396262 0.64166476
[805] 0.09874672 0.74117230 0.88042303 0.20463527 0.07716103 0.76130711
[811] 0.35868103 0.21040180 0.39508843 0.55011624 0.62901754 0.07691866
[817] 0.15624001 0.09967459 0.45430012 0.30479252 0.29333327 0.98514343
[823] 0.66769342 0.24326049 0.36234112 0.91644893 0.98642841 0.93727508
[829] 0.47331589 0.72444770 0.97023652 0.71696026 0.05951111 0.79186299
[835] 0.44727789 0.13038835 0.24027433 0.24880052 0.17589295 0.29718505
[841] 0.57942784 0.38394402 0.74071859 0.08950177 0.92844995 0.20546273
[847] 0.17052521 0.56556488 0.53522199 0.65044850 0.62522401 0.84069901
[853] 0.30333271 0.47058932 0.74251430 0.75843285 0.78986565 0.64000091
[859] 0.95974780 0.55839972 0.15998748 0.49666129 0.76857653 0.10026845
[865] 0.64366779 0.44003174 0.41981984 0.06424624 0.37838196 0.65536704
[871] 0.49158512 0.95845727 0.82220819 0.71594727 0.15195111 0.87880472
[877] 0.56239340 0.75575607 0.66005890 0.10042827 0.07476948 0.68994095
[883] 0.53777551 0.89742428 0.48578599 0.53196026 0.80053556 0.33327260
[889] 0.28729421 0.87427012 0.30134071 0.93290339 0.86937539 0.96247036
[895] 0.69094520 0.87007512 0.59991133 0.05653677 0.56845776 0.85619272
[901] 0.51146998 0.35123484 0.49837745 0.49197291 0.31215888 0.13636030
[907] 0.88629032 0.35219993 0.28057341 0.67752457 0.47881093 0.88306662
[913] 0.18275772 0.42877247 0.35729102 0.62259801 0.86923170 0.38988416
[919] 0.86267826 0.14169013 0.16083008 0.45782094 0.07666457 0.19357154
[925] 0.61379033 0.77079887 0.35561980 0.82658629 0.23974572 0.09301519
[931] 0.28104933 0.74090923 0.55055417 0.44111115 0.34152779 0.26156970
[937] 0.67222906 0.08931934 0.54642699 0.60435492 0.67172228 0.98392426
[943] 0.35895814 0.54127555 0.99988300 0.98867658 0.20519919 0.70817382
b.
P=c()
for ( i in 1:1000){
A=rnorm(5, 8.4,0.05)
B=rnorm(5, 8.4,0.05)
C=rnorm(5, 8.45,0.05)
Y=c(A,B,C)
X=gl(3,5)
M=model.matrix(Y ~ X)
summary( lm(Y~X))
P[i]=anova(lm(Y~X))$Pr[1]
}
PF=P[P<0.05]
[1] 5.985202e-04 1.583619e-02 2.738370e-02 1.292017e-03 3.265909e-02
[6] 1.531942e-02 3.621216e-02 1.979093e-02 9.602396e-03 2.059242e-02
[11] 2.395804e-02 3.168572e-02 3.115089e-02 3.570280e-02 3.659052e-02
[16] 3.738629e-02 2.869078e-02 1.633252e-02 2.979475e-02 1.050423e-02
[21] 4.084896e-03 3.271097e-02 4.889220e-02 1.158635e-02 2.616998e-02
[26] 2.857227e-02 3.746069e-02 4.497941e-02 4.283319e-02 1.236978e-02
[31] 6.143888e-04 1.476580e-02 2.990340e-02 4.048845e-02 2.744873e-02
[36] 4.091743e-03 1.410260e-02 3.133262e-02 1.197649e-02 2.413866e-03
[41] 4.083580e-02 2.287754e-03 1.842255e-02 7.408790e-03 4.988074e-02
[46] 1.388203e-02 3.822465e-02 3.702682e-03 1.199860e-02 1.860195e-02
[51] 2.880912e-02 3.731410e-02 1.268287e-02 9.549247e-03 3.664141e-03
[56] 1.632231e-02 1.381114e-02 6.344002e-03 6.905466e-03 2.902254e-02
[61] 3.248156e-02 3.457028e-02 4.145729e-02 3.183430e-02 1.167909e-03
[66] 9.738157e-03 3.086900e-02 2.176421e-02 9.644607e-03 3.577540e-02
[71] 5.664481e-03 2.702272e-02 3.261639e-02 1.944421e-02 4.666645e-02
[76] 1.685214e-02 2.991360e-03 3.302672e-03 2.261474e-02 4.537402e-02
[81] 1.902533e-02 1.645822e-02 4.435867e-03 3.390808e-02 4.171277e-02
[86] 2.049796e-02 2.066172e-02 3.956197e-02 2.997112e-03 1.542726e-02
[91] 3.392757e-02 1.677382e-02 3.732427e-02 5.816145e-03 2.029523e-02
[96] 4.757786e-02 3.780963e-02 1.616493e-03 1.129113e-03 4.673321e-03
[101] 1.408720e-02 7.943327e-03 2.710843e-03 4.056892e-04 4.625924e-02
[106] 2.660587e-02 2.891661e-02 1.146063e-03 3.944798e-04 2.352799e-02
[111] 2.262285e-02 2.961109e-02 5.543839e-04 1.149149e-02 3.363004e-02
[116] 2.798538e-02 1.949266e-04 1.808396e-04 2.502964e-02 7.393068e-03
[121] 1.319852e-02 3.759907e-03 3.606385e-02 4.327140e-03 1.992402e-02
[126] 2.760056e-03 1.924673e-02 3.634359e-02 1.224644e-02 3.087496e-02
[131] 4.377843e-02 1.061089e-02 1.976468e-03 2.269922e-03 4.595414e-02
[136] 1.020181e-02 2.577209e-02 2.833687e-02 9.118215e-03 9.252963e-03
[141] 1.383517e-02 3.196889e-02 1.365600e-02 1.566171e-02 2.899789e-02
[146] 3.603833e-02 1.530470e-02 2.394644e-02 4.759710e-03 1.930537e-02
[151] 1.404394e-02 1.614801e-02 9.766626e-04 9.874560e-04 2.813266e-02
[156] 4.290957e-03 4.169290e-03 3.232997e-02 1.569108e-03 2.610755e-02
[161] 4.945182e-02 2.192871e-02 4.913319e-02 6.303048e-04 5.790629e-03
[166] 4.803617e-02 4.096355e-02 1.508932e-02 3.006727e-02 4.741326e-02
[171] 1.124614e-02 9.647569e-03 2.547173e-02 2.664509e-02 2.197237e-02
[176] 1.501341e-02 3.148551e-02 1.508757e-02 1.836874e-02 3.818659e-02
[181] 1.051707e-02 1.065644e-02 5.215832e-03 5.405680e-03 3.573474e-02
[186] 1.976329e-03 3.682097e-02 4.068190e-02 2.368561e-02 1.974915e-02
[191] 9.782351e-03 6.280924e-03 4.096486e-02 4.805602e-03 2.474576e-02
[196] 8.657223e-03 2.080517e-02 1.149807e-02 2.973402e-02 4.608079e-02
[201] 2.573053e-02 1.135367e-02 4.712994e-02 3.369725e-03 1.158359e-02
[206] 4.695125e-02 3.446343e-03 1.061459e-02 2.195110e-02 1.116056e-02
[211] 3.617409e-02 3.081532e-03 5.472311e-03 1.955048e-02 2.244265e-03
[216] 4.159821e-02 3.386046e-02 4.671594e-03 9.378109e-03 4.901152e-02
[221] 1.196697e-02 3.047246e-02 1.454892e-02 4.546986e-02 3.606603e-02
[226] 1.090744e-03 4.821838e-02 4.260064e-03 3.974703e-03 2.296563e-03
[231] 4.177295e-02 1.022631e-02 1.990582e-02 8.525393e-03 2.522714e-02
[236] 4.783689e-02 1.974644e-02 2.671705e-02 4.437788e-02 1.938711e-02
[241] 9.225025e-03 2.658876e-02 3.667207e-05 1.684434e-02 1.163043e-02
[246] 3.472581e-02 3.167938e-03 2.924813e-02 5.666923e-03 1.329268e-02
[251] 3.982673e-02 2.087590e-02 4.272664e-02 5.509284e-05 4.302013e-02
[256] 7.810956e-03 3.596718e-02 4.650805e-03 1.799112e-03 3.293295e-03
[261] 4.274090e-02 4.702361e-02 6.461186e-03 4.407187e-02 4.145870e-02
[266] 2.902578e-02 1.389830e-02 1.307531e-02 1.632320e-02 4.626143e-02
[271] 4.565935e-02 5.927693e-03 1.485215e-02 7.712736e-03 1.002472e-02
[276] 2.047561e-04 3.690436e-02 2.501171e-02 4.948299e-02 2.278222e-02
[281] 2.160229e-02 2.740538e-02 3.677766e-03 1.841639e-02 4.806974e-04
[286] 3.776953e-02 1.607549e-02 7.718202e-04 1.048345e-03 1.801317e-03
[291] 1.660349e-02 5.343627e-03 7.578584e-04 4.572279e-02 3.033498e-02
[296] 1.265506e-02