Atividade 7
Isabelle Ferreira
2023-05-25
Atividade - Probabilidade Normal
Introdução
Este trabalho fará os cálculos das probabilidades apresentadas abaixo de uma distribuição normal.
Carregando os pacotes
library(RcmdrMisc)
library(dplyr)Fatores
x = seq(-3.291, 3.291, length.out=1000)
x## [1] -3.291000000 -3.284411411 -3.277822823 -3.271234234 -3.264645646
## [6] -3.258057057 -3.251468468 -3.244879880 -3.238291291 -3.231702703
## [11] -3.225114114 -3.218525526 -3.211936937 -3.205348348 -3.198759760
## [16] -3.192171171 -3.185582583 -3.178993994 -3.172405405 -3.165816817
## [21] -3.159228228 -3.152639640 -3.146051051 -3.139462462 -3.132873874
## [26] -3.126285285 -3.119696697 -3.113108108 -3.106519520 -3.099930931
## [31] -3.093342342 -3.086753754 -3.080165165 -3.073576577 -3.066987988
## [36] -3.060399399 -3.053810811 -3.047222222 -3.040633634 -3.034045045
## [41] -3.027456456 -3.020867868 -3.014279279 -3.007690691 -3.001102102
## [46] -2.994513514 -2.987924925 -2.981336336 -2.974747748 -2.968159159
## [51] -2.961570571 -2.954981982 -2.948393393 -2.941804805 -2.935216216
## [56] -2.928627628 -2.922039039 -2.915450450 -2.908861862 -2.902273273
## [61] -2.895684685 -2.889096096 -2.882507508 -2.875918919 -2.869330330
## [66] -2.862741742 -2.856153153 -2.849564565 -2.842975976 -2.836387387
## [71] -2.829798799 -2.823210210 -2.816621622 -2.810033033 -2.803444444
## [76] -2.796855856 -2.790267267 -2.783678679 -2.777090090 -2.770501502
## [81] -2.763912913 -2.757324324 -2.750735736 -2.744147147 -2.737558559
## [86] -2.730969970 -2.724381381 -2.717792793 -2.711204204 -2.704615616
## [91] -2.698027027 -2.691438438 -2.684849850 -2.678261261 -2.671672673
## [96] -2.665084084 -2.658495495 -2.651906907 -2.645318318 -2.638729730
## [101] -2.632141141 -2.625552553 -2.618963964 -2.612375375 -2.605786787
## [106] -2.599198198 -2.592609610 -2.586021021 -2.579432432 -2.572843844
## [111] -2.566255255 -2.559666667 -2.553078078 -2.546489489 -2.539900901
## [116] -2.533312312 -2.526723724 -2.520135135 -2.513546547 -2.506957958
## [121] -2.500369369 -2.493780781 -2.487192192 -2.480603604 -2.474015015
## [126] -2.467426426 -2.460837838 -2.454249249 -2.447660661 -2.441072072
## [131] -2.434483483 -2.427894895 -2.421306306 -2.414717718 -2.408129129
## [136] -2.401540541 -2.394951952 -2.388363363 -2.381774775 -2.375186186
## [141] -2.368597598 -2.362009009 -2.355420420 -2.348831832 -2.342243243
## [146] -2.335654655 -2.329066066 -2.322477477 -2.315888889 -2.309300300
## [151] -2.302711712 -2.296123123 -2.289534535 -2.282945946 -2.276357357
## [156] -2.269768769 -2.263180180 -2.256591592 -2.250003003 -2.243414414
## [161] -2.236825826 -2.230237237 -2.223648649 -2.217060060 -2.210471471
## [166] -2.203882883 -2.197294294 -2.190705706 -2.184117117 -2.177528529
## [171] -2.170939940 -2.164351351 -2.157762763 -2.151174174 -2.144585586
## [176] -2.137996997 -2.131408408 -2.124819820 -2.118231231 -2.111642643
## [181] -2.105054054 -2.098465465 -2.091876877 -2.085288288 -2.078699700
## [186] -2.072111111 -2.065522523 -2.058933934 -2.052345345 -2.045756757
## [191] -2.039168168 -2.032579580 -2.025990991 -2.019402402 -2.012813814
## [196] -2.006225225 -1.999636637 -1.993048048 -1.986459459 -1.979870871
## [201] -1.973282282 -1.966693694 -1.960105105 -1.953516517 -1.946927928
## [206] -1.940339339 -1.933750751 -1.927162162 -1.920573574 -1.913984985
## [211] -1.907396396 -1.900807808 -1.894219219 -1.887630631 -1.881042042
## [216] -1.874453453 -1.867864865 -1.861276276 -1.854687688 -1.848099099
## [221] -1.841510511 -1.834921922 -1.828333333 -1.821744745 -1.815156156
## [226] -1.808567568 -1.801978979 -1.795390390 -1.788801802 -1.782213213
## [231] -1.775624625 -1.769036036 -1.762447447 -1.755858859 -1.749270270
## [236] -1.742681682 -1.736093093 -1.729504505 -1.722915916 -1.716327327
## [241] -1.709738739 -1.703150150 -1.696561562 -1.689972973 -1.683384384
## [246] -1.676795796 -1.670207207 -1.663618619 -1.657030030 -1.650441441
## [251] -1.643852853 -1.637264264 -1.630675676 -1.624087087 -1.617498498
## [256] -1.610909910 -1.604321321 -1.597732733 -1.591144144 -1.584555556
## [261] -1.577966967 -1.571378378 -1.564789790 -1.558201201 -1.551612613
## [266] -1.545024024 -1.538435435 -1.531846847 -1.525258258 -1.518669670
## [271] -1.512081081 -1.505492492 -1.498903904 -1.492315315 -1.485726727
## [276] -1.479138138 -1.472549550 -1.465960961 -1.459372372 -1.452783784
## [281] -1.446195195 -1.439606607 -1.433018018 -1.426429429 -1.419840841
## [286] -1.413252252 -1.406663664 -1.400075075 -1.393486486 -1.386897898
## [291] -1.380309309 -1.373720721 -1.367132132 -1.360543544 -1.353954955
## [296] -1.347366366 -1.340777778 -1.334189189 -1.327600601 -1.321012012
## [301] -1.314423423 -1.307834835 -1.301246246 -1.294657658 -1.288069069
## [306] -1.281480480 -1.274891892 -1.268303303 -1.261714715 -1.255126126
## [311] -1.248537538 -1.241948949 -1.235360360 -1.228771772 -1.222183183
## [316] -1.215594595 -1.209006006 -1.202417417 -1.195828829 -1.189240240
## [321] -1.182651652 -1.176063063 -1.169474474 -1.162885886 -1.156297297
## [326] -1.149708709 -1.143120120 -1.136531532 -1.129942943 -1.123354354
## [331] -1.116765766 -1.110177177 -1.103588589 -1.097000000 -1.090411411
## [336] -1.083822823 -1.077234234 -1.070645646 -1.064057057 -1.057468468
## [341] -1.050879880 -1.044291291 -1.037702703 -1.031114114 -1.024525526
## [346] -1.017936937 -1.011348348 -1.004759760 -0.998171171 -0.991582583
## [351] -0.984993994 -0.978405405 -0.971816817 -0.965228228 -0.958639640
## [356] -0.952051051 -0.945462462 -0.938873874 -0.932285285 -0.925696697
## [361] -0.919108108 -0.912519520 -0.905930931 -0.899342342 -0.892753754
## [366] -0.886165165 -0.879576577 -0.872987988 -0.866399399 -0.859810811
## [371] -0.853222222 -0.846633634 -0.840045045 -0.833456456 -0.826867868
## [376] -0.820279279 -0.813690691 -0.807102102 -0.800513514 -0.793924925
## [381] -0.787336336 -0.780747748 -0.774159159 -0.767570571 -0.760981982
## [386] -0.754393393 -0.747804805 -0.741216216 -0.734627628 -0.728039039
## [391] -0.721450450 -0.714861862 -0.708273273 -0.701684685 -0.695096096
## [396] -0.688507508 -0.681918919 -0.675330330 -0.668741742 -0.662153153
## [401] -0.655564565 -0.648975976 -0.642387387 -0.635798799 -0.629210210
## [406] -0.622621622 -0.616033033 -0.609444444 -0.602855856 -0.596267267
## [411] -0.589678679 -0.583090090 -0.576501502 -0.569912913 -0.563324324
## [416] -0.556735736 -0.550147147 -0.543558559 -0.536969970 -0.530381381
## [421] -0.523792793 -0.517204204 -0.510615616 -0.504027027 -0.497438438
## [426] -0.490849850 -0.484261261 -0.477672673 -0.471084084 -0.464495495
## [431] -0.457906907 -0.451318318 -0.444729730 -0.438141141 -0.431552553
## [436] -0.424963964 -0.418375375 -0.411786787 -0.405198198 -0.398609610
## [441] -0.392021021 -0.385432432 -0.378843844 -0.372255255 -0.365666667
## [446] -0.359078078 -0.352489489 -0.345900901 -0.339312312 -0.332723724
## [451] -0.326135135 -0.319546547 -0.312957958 -0.306369369 -0.299780781
## [456] -0.293192192 -0.286603604 -0.280015015 -0.273426426 -0.266837838
## [461] -0.260249249 -0.253660661 -0.247072072 -0.240483483 -0.233894895
## [466] -0.227306306 -0.220717718 -0.214129129 -0.207540541 -0.200951952
## [471] -0.194363363 -0.187774775 -0.181186186 -0.174597598 -0.168009009
## [476] -0.161420420 -0.154831832 -0.148243243 -0.141654655 -0.135066066
## [481] -0.128477477 -0.121888889 -0.115300300 -0.108711712 -0.102123123
## [486] -0.095534535 -0.088945946 -0.082357357 -0.075768769 -0.069180180
## [491] -0.062591592 -0.056003003 -0.049414414 -0.042825826 -0.036237237
## [496] -0.029648649 -0.023060060 -0.016471471 -0.009882883 -0.003294294
## [501] 0.003294294 0.009882883 0.016471471 0.023060060 0.029648649
## [506] 0.036237237 0.042825826 0.049414414 0.056003003 0.062591592
## [511] 0.069180180 0.075768769 0.082357357 0.088945946 0.095534535
## [516] 0.102123123 0.108711712 0.115300300 0.121888889 0.128477477
## [521] 0.135066066 0.141654655 0.148243243 0.154831832 0.161420420
## [526] 0.168009009 0.174597598 0.181186186 0.187774775 0.194363363
## [531] 0.200951952 0.207540541 0.214129129 0.220717718 0.227306306
## [536] 0.233894895 0.240483483 0.247072072 0.253660661 0.260249249
## [541] 0.266837838 0.273426426 0.280015015 0.286603604 0.293192192
## [546] 0.299780781 0.306369369 0.312957958 0.319546547 0.326135135
## [551] 0.332723724 0.339312312 0.345900901 0.352489489 0.359078078
## [556] 0.365666667 0.372255255 0.378843844 0.385432432 0.392021021
## [561] 0.398609610 0.405198198 0.411786787 0.418375375 0.424963964
## [566] 0.431552553 0.438141141 0.444729730 0.451318318 0.457906907
## [571] 0.464495495 0.471084084 0.477672673 0.484261261 0.490849850
## [576] 0.497438438 0.504027027 0.510615616 0.517204204 0.523792793
## [581] 0.530381381 0.536969970 0.543558559 0.550147147 0.556735736
## [586] 0.563324324 0.569912913 0.576501502 0.583090090 0.589678679
## [591] 0.596267267 0.602855856 0.609444444 0.616033033 0.622621622
## [596] 0.629210210 0.635798799 0.642387387 0.648975976 0.655564565
## [601] 0.662153153 0.668741742 0.675330330 0.681918919 0.688507508
## [606] 0.695096096 0.701684685 0.708273273 0.714861862 0.721450450
## [611] 0.728039039 0.734627628 0.741216216 0.747804805 0.754393393
## [616] 0.760981982 0.767570571 0.774159159 0.780747748 0.787336336
## [621] 0.793924925 0.800513514 0.807102102 0.813690691 0.820279279
## [626] 0.826867868 0.833456456 0.840045045 0.846633634 0.853222222
## [631] 0.859810811 0.866399399 0.872987988 0.879576577 0.886165165
## [636] 0.892753754 0.899342342 0.905930931 0.912519520 0.919108108
## [641] 0.925696697 0.932285285 0.938873874 0.945462462 0.952051051
## [646] 0.958639640 0.965228228 0.971816817 0.978405405 0.984993994
## [651] 0.991582583 0.998171171 1.004759760 1.011348348 1.017936937
## [656] 1.024525526 1.031114114 1.037702703 1.044291291 1.050879880
## [661] 1.057468468 1.064057057 1.070645646 1.077234234 1.083822823
## [666] 1.090411411 1.097000000 1.103588589 1.110177177 1.116765766
## [671] 1.123354354 1.129942943 1.136531532 1.143120120 1.149708709
## [676] 1.156297297 1.162885886 1.169474474 1.176063063 1.182651652
## [681] 1.189240240 1.195828829 1.202417417 1.209006006 1.215594595
## [686] 1.222183183 1.228771772 1.235360360 1.241948949 1.248537538
## [691] 1.255126126 1.261714715 1.268303303 1.274891892 1.281480480
## [696] 1.288069069 1.294657658 1.301246246 1.307834835 1.314423423
## [701] 1.321012012 1.327600601 1.334189189 1.340777778 1.347366366
## [706] 1.353954955 1.360543544 1.367132132 1.373720721 1.380309309
## [711] 1.386897898 1.393486486 1.400075075 1.406663664 1.413252252
## [716] 1.419840841 1.426429429 1.433018018 1.439606607 1.446195195
## [721] 1.452783784 1.459372372 1.465960961 1.472549550 1.479138138
## [726] 1.485726727 1.492315315 1.498903904 1.505492492 1.512081081
## [731] 1.518669670 1.525258258 1.531846847 1.538435435 1.545024024
## [736] 1.551612613 1.558201201 1.564789790 1.571378378 1.577966967
## [741] 1.584555556 1.591144144 1.597732733 1.604321321 1.610909910
## [746] 1.617498498 1.624087087 1.630675676 1.637264264 1.643852853
## [751] 1.650441441 1.657030030 1.663618619 1.670207207 1.676795796
## [756] 1.683384384 1.689972973 1.696561562 1.703150150 1.709738739
## [761] 1.716327327 1.722915916 1.729504505 1.736093093 1.742681682
## [766] 1.749270270 1.755858859 1.762447447 1.769036036 1.775624625
## [771] 1.782213213 1.788801802 1.795390390 1.801978979 1.808567568
## [776] 1.815156156 1.821744745 1.828333333 1.834921922 1.841510511
## [781] 1.848099099 1.854687688 1.861276276 1.867864865 1.874453453
## [786] 1.881042042 1.887630631 1.894219219 1.900807808 1.907396396
## [791] 1.913984985 1.920573574 1.927162162 1.933750751 1.940339339
## [796] 1.946927928 1.953516517 1.960105105 1.966693694 1.973282282
## [801] 1.979870871 1.986459459 1.993048048 1.999636637 2.006225225
## [806] 2.012813814 2.019402402 2.025990991 2.032579580 2.039168168
## [811] 2.045756757 2.052345345 2.058933934 2.065522523 2.072111111
## [816] 2.078699700 2.085288288 2.091876877 2.098465465 2.105054054
## [821] 2.111642643 2.118231231 2.124819820 2.131408408 2.137996997
## [826] 2.144585586 2.151174174 2.157762763 2.164351351 2.170939940
## [831] 2.177528529 2.184117117 2.190705706 2.197294294 2.203882883
## [836] 2.210471471 2.217060060 2.223648649 2.230237237 2.236825826
## [841] 2.243414414 2.250003003 2.256591592 2.263180180 2.269768769
## [846] 2.276357357 2.282945946 2.289534535 2.296123123 2.302711712
## [851] 2.309300300 2.315888889 2.322477477 2.329066066 2.335654655
## [856] 2.342243243 2.348831832 2.355420420 2.362009009 2.368597598
## [861] 2.375186186 2.381774775 2.388363363 2.394951952 2.401540541
## [866] 2.408129129 2.414717718 2.421306306 2.427894895 2.434483483
## [871] 2.441072072 2.447660661 2.454249249 2.460837838 2.467426426
## [876] 2.474015015 2.480603604 2.487192192 2.493780781 2.500369369
## [881] 2.506957958 2.513546547 2.520135135 2.526723724 2.533312312
## [886] 2.539900901 2.546489489 2.553078078 2.559666667 2.566255255
## [891] 2.572843844 2.579432432 2.586021021 2.592609610 2.599198198
## [896] 2.605786787 2.612375375 2.618963964 2.625552553 2.632141141
## [901] 2.638729730 2.645318318 2.651906907 2.658495495 2.665084084
## [906] 2.671672673 2.678261261 2.684849850 2.691438438 2.698027027
## [911] 2.704615616 2.711204204 2.717792793 2.724381381 2.730969970
## [916] 2.737558559 2.744147147 2.750735736 2.757324324 2.763912913
## [921] 2.770501502 2.777090090 2.783678679 2.790267267 2.796855856
## [926] 2.803444444 2.810033033 2.816621622 2.823210210 2.829798799
## [931] 2.836387387 2.842975976 2.849564565 2.856153153 2.862741742
## [936] 2.869330330 2.875918919 2.882507508 2.889096096 2.895684685
## [941] 2.902273273 2.908861862 2.915450450 2.922039039 2.928627628
## [946] 2.935216216 2.941804805 2.948393393 2.954981982 2.961570571
## [951] 2.968159159 2.974747748 2.981336336 2.987924925 2.994513514
## [956] 3.001102102 3.007690691 3.014279279 3.020867868 3.027456456
## [961] 3.034045045 3.040633634 3.047222222 3.053810811 3.060399399
## [966] 3.066987988 3.073576577 3.080165165 3.086753754 3.093342342
## [971] 3.099930931 3.106519520 3.113108108 3.119696697 3.126285285
## [976] 3.132873874 3.139462462 3.146051051 3.152639640 3.159228228
## [981] 3.165816817 3.172405405 3.178993994 3.185582583 3.192171171
## [986] 3.198759760 3.205348348 3.211936937 3.218525526 3.225114114
## [991] 3.231702703 3.238291291 3.244879880 3.251468468 3.258057057
## [996] 3.264645646 3.271234234 3.277822823 3.284411411 3.2910000001º
X ~ N (50,30)
A média é 50 e o desvio padrão é 30.
a)
Pr(X<50) =
Pr(X-media)/sigma < (50 - 50)/30 =
Pr(Z<-0/30) = Pr(z<0) = 0.5 - Pr(0<Z<0) =
0.5 - Pr(0<Z<0) = 0.5 - 0.01994 = 0.48006
pnorm(0)
pnorm(50, mean = 50, sd=30)
plotDistr(x, dnorm(x, mean=0, sd=1), cdf=FALSE, xlab="x",
ylab="Densidade",regions=list(c(-10,0), c(0,10)),
col=c('#78b8de', '#f2dc8d'), legend=FALSE)
b)
Pr(40<X<61) =
Pr[(40 - 50)/30<Z<(61-50/30)] =
Pr(-0,33<Z<0,37) = Pr(0<Z<0,33) + Pr(0<Z<0,37) =
pnorm(0.37) - pnorm(-0.33) = 0.2736088
plotDistr(x, dnorm(x, mean=0, sd=1), cdf=FALSE, xlab="x",
ylab="Densidade",regions=list(c(-0.33, 0),c(0, 0.37)),
col=c('#f2a16b', '#b2edbc'), legend=FALSE)
c)
Pr(33<X<47) =
Pr[(33 - 50)/30<Z<(47-50/30)]
Pr(-0.57<Z<-0.1) = Pr(0<Z<0.57) + Pr(0<Z<0.1) =
pnorm(0.57) - pnorm(0.1) = 0.1758333
plotDistr(x, dnorm(x, mean=0, sd=1), cdf=FALSE, xlab="x",
ylab="Densidade",regions=list(c(0, 0.57),c(0.1, 0.57)),
col=c('#c0ddeb', '#f5a4db'), legend=FALSE)
2º
Y ~ N(10,15)
A média é 10 e o desvio padrão é 15.
a)
Pr(5<Y<7) =
Pr[(5 - 10)/15<Z<(7-10/15)] =
Pr(-0.33<Z<-0.2) = Pr(0<Z<0.33) + Pr(0<Z<0.2) =
pnorm(0.33) - pnorm(0.2) = 0.05004031
plotDistr(x, dnorm(x, mean=0, sd=1), cdf=FALSE, xlab="x",
ylab="Densidade",regions=list(c(0, 0.2),c(0.2, 0.33)),
col=c('#62f075', '#db8f88'), legend=FALSE)
b)
Pr(9.5<Y<11) =
Pr[(9.5 - 10)/15<Z<(11-10/15)] =
Pr(-0.03<Z<0.06) = Pr(0<Z<0.03) + Pr(0<Z<0.06) =
pnorm(0.06) - pnorm(-0.3) = 0.1418336
plotDistr(x, dnorm(x, mean=0, sd=1), cdf=FALSE, xlab="x",
ylab="Densidade",regions=list(c(-0.03, 0),c(0, 0.06)),
col=c('#babd13', '#5475eb'), legend=FALSE)
c)
Pr(Y>7.5) =
Pr(X-media)/sigma > (10 - 7.5)/15 =
Pr(Z>2.5/15) = Pr(z>0.16) = (0.16>Z>0) =
pnorm(0.16) - pnorm(0) = 0.06355946
plotDistr(x, dnorm(x, mean=0, sd=1), cdf=FALSE, xlab="x",
ylab="Densidade",regions=list(c(0, 0.16)),
col=c('#37e697', 'red'), legend=FALSE)
Conclusão
Concluo minha 7ª atividade de estatística após redigir este documento com todos os cálculos propostos.