Problema 1.

Tenemos en cuenta los percentiles para establecer los límites.

datos=Capability_CD$`CD Thickness (mm)`
summary(datos)
##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##   1.188   1.313   1.349   1.349   1.389   1.486
lsl=1.313
usl=1.389

Una vez tenemos los límites superior e inferior ajustamos el análisis para ver si los datos cumplen los parámetros estabecidos.

#ejercicio base
ss.study.ca(xST=datos,Target = mean(datos),LSL = lsl, USL = usl, f.sub = "original")

#Comparación corto a largo plazo
ss.study.ca(xST=datos[1:25],xLT = datos[26:100],Target = mean(datos),LSL = lsl, USL = usl, f.sub = "original largo plazo")

#Nº defectos del análisis
ss.ca.yield(defects = sum(datos<lsl | datos>usl), rework = 0, opportunities = 100)
##   Yield  FTY  RTY DPU   DPMO
## 1  0.52 0.52 0.52  48 480000

Como podemos ver el análisis con los cuartiles como intervalo está centrado pero no es adecuado, debido a que el índice de capacidad del proceso(Cp) y el índice de capacidad ajustado(Cpk) esinferior a 1.

Calculamos los nuevos límites aplicando el índice intercuartílico y analizamos los nuevos resultados.

boxplot(datos)

lsl2=quantile(datos,c(0.25))-IQR(datos)*1.5
usl2=quantile(datos,c(0.75))+IQR(datos)*1.5
#ejercicio base mejorado
ss.study.ca(xST=datos,T = mean(datos),LSL = lsl2, USL = usl2, f.sub = "mejorado")

#Comparación corto a largo plazo
ss.study.ca(xST=datos[1:25],xLT = datos[26:100],Target = mean(datos),LSL = lsl2, USL = usl2, f.sub = "mejorado largo plazo")

#Nº defectos del análisis
ss.ca.yield(defects = sum(datos<lsl2 | datos>usl2), rework = 0, opportunities = 100)
##   Yield  FTY  RTY DPU  DPMO
## 1  0.99 0.99 0.99   1 10000

Con los nuevos límites ahora todos los datos están dentro de los límites de especificación excepto uno. El problema mejora pero el análisis sigue siendo inadecuado.

Problema 2.

Al igual que al problema 1 tomamos los cuartiles

datos2=Capability_RepairTimes$`Time to fix fault (decimal hrs)`
summary(datos2)
##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##   3.420   4.515   5.125   5.033   5.530   6.610
lsl3=4.515
usl3=5.530

Tenemos en cuenta los datos sin importar a que subgrupo pertenece.

#ejercicio base
ss.study.ca(xST=datos2,Target = mean(datos2),LSL = lsl3, USL = usl3, f.sub = "original")

#Comparación corto a largo plazo
ss.study.ca(xST=datos2[1:25],xLT = datos2[26:100],Target = mean(datos2),LSL = lsl3, USL = usl3, f.sub = "original largo plazo")

#Nº defectos del análisis
ss.ca.yield(defects = sum(datos2<lsl3 | datos2>usl3), rework = 0, opportunities = 100)
##   Yield FTY RTY DPU  DPMO
## 1   0.5 0.5 0.5  50 5e+05

En este caso el primer apartado también está centrado como en el ejercicio anterior pero a largo plazo los datos están descentrados. En ambos casos los índices nos siguen dando valores poco deseados.

Volvemos a repetir el procedimiento pero con los nuevos límites.

boxplot(datos2)

lsl4 = quantile(datos2,c(0.25))-IQR(datos2)*1.5
usl4 = quantile(datos2,c(0.75))+IQR(datos2)*1.5
#ejercicio base mejorado
ss.study.ca(xST=datos2,T = mean(datos2),LSL = lsl4, USL = usl4, f.sub = "mejorado")

#Comparación corto a largo plazo
ss.study.ca(xST=datos2[1:25],xLT = datos2[26:100],Target = mean(datos2),LSL = lsl4, USL = usl4, f.sub = "mejorado largo plazo")

#Nº defectos del análisis
ss.ca.yield(defects = sum(datos2<lsl4 | datos2>usl4), rework = 0, opportunities = 100)
##   Yield FTY RTY DPU DPMO
## 1     1   1   1   0    0

Sigue estando descentrado a largo plazo pero ahora el análisis es adecuado.

Para responder a la pregunta de si hay diferencia entre los grupos hemos realizado un ánalisis anova y un test de HSD para comparar los grupos, finalmente hemos sacado la media, mínima y máxima de cada grupo para ver las diferencias.

datos3=Capability_RepairTimes
names(datos3)=c("tiempo","grupos")

attach(datos3)
fm= aov(lm(tiempo~as.factor(grupos)))
summary(fm)
##                   Df Sum Sq Mean Sq F value Pr(>F)    
## as.factor(grupos) 19  43.78   2.304   12.32 <2e-16 ***
## Residuals         80  14.96   0.187                   
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
TukeyHSD(fm)
##   Tukey multiple comparisons of means
##     95% family-wise confidence level
## 
## Fit: aov(formula = lm(tiempo ~ as.factor(grupos)))
## 
## $`as.factor(grupos)`
##         diff          lwr          upr     p adj
## 2-1   -0.352 -1.354467805  0.650467805 0.9989781
## 3-1    0.190 -0.812467805  1.192467805 0.9999999
## 4-1   -0.024 -1.026467805  0.978467805 1.0000000
## 5-1    0.588 -0.414467805  1.590467805 0.8203962
## 6-1    0.506 -0.496467805  1.508467805 0.9434921
## 7-1    0.410 -0.592467805  1.412467805 0.9933266
## 8-1    0.214 -0.788467805  1.216467805 0.9999994
## 9-1    0.568 -0.434467805  1.570467805 0.8583786
## 10-1   0.110 -0.892467805  1.112467805 1.0000000
## 11-1  -0.466 -1.468467805  0.536467805 0.9737868
## 12-1  -0.944 -1.946467805  0.058467805 0.0897045
## 13-1  -0.596 -1.598467805  0.406467805 0.8038692
## 14-1  -0.888 -1.890467805  0.114467805 0.1495089
## 15-1  -1.288 -2.290467805 -0.285532195 0.0016119
## 16-1  -1.194 -2.196467805 -0.191532195 0.0054617
## 17-1  -1.336 -2.338467805 -0.333532195 0.0008412
## 18-1  -1.516 -2.518467805 -0.513532195 0.0000643
## 19-1  -0.200 -1.202467805  0.802467805 0.9999998
## 20-1  -0.476 -1.478467805  0.526467805 0.9677660
## 3-2    0.542 -0.460467805  1.544467805 0.9001451
## 4-2    0.328 -0.674467805  1.330467805 0.9996040
## 5-2    0.940 -0.062467805  1.942467805 0.0931923
## 6-2    0.858 -0.144467805  1.860467805 0.1923584
## 7-2    0.762 -0.240467805  1.764467805 0.3846573
## 8-2    0.566 -0.436467805  1.568467805 0.8619027
## 9-2    0.920 -0.082467805  1.922467805 0.1123528
## 10-2   0.462 -0.540467805  1.464467805 0.9759383
## 11-2  -0.114 -1.116467805  0.888467805 1.0000000
## 12-2  -0.592 -1.594467805  0.410467805 0.8122238
## 13-2  -0.244 -1.246467805  0.758467805 0.9999952
## 14-2  -0.536 -1.538467805  0.466467805 0.9085264
## 15-2  -0.936 -1.938467805  0.066467805 0.0967918
## 16-2  -0.842 -1.844467805  0.160467805 0.2185857
## 17-2  -0.984 -1.986467805  0.018467805 0.0604573
## 18-2  -1.164 -2.166467805 -0.161532195 0.0079302
## 19-2   0.152 -0.850467805  1.154467805 1.0000000
## 20-2  -0.124 -1.126467805  0.878467805 1.0000000
## 4-3   -0.214 -1.216467805  0.788467805 0.9999994
## 5-3    0.398 -0.604467805  1.400467805 0.9952766
## 6-3    0.316 -0.686467805  1.318467805 0.9997642
## 7-3    0.220 -0.782467805  1.222467805 0.9999991
## 8-3    0.024 -0.978467805  1.026467805 1.0000000
## 9-3    0.378 -0.624467805  1.380467805 0.9974673
## 10-3  -0.080 -1.082467805  0.922467805 1.0000000
## 11-3  -0.656 -1.658467805  0.346467805 0.6607058
## 12-3  -1.134 -2.136467805 -0.131532195 0.0114131
## 13-3  -0.786 -1.788467805  0.216467805 0.3290603
## 14-3  -1.078 -2.080467805 -0.075532195 0.0219501
## 15-3  -1.478 -2.480467805 -0.475532195 0.0001124
## 16-3  -1.384 -2.386467805 -0.381532195 0.0004320
## 17-3  -1.526 -2.528467805 -0.523532195 0.0000555
## 18-3  -1.706 -2.708467805 -0.703532195 0.0000036
## 19-3  -0.390 -1.392467805  0.612467805 0.9962917
## 20-3  -0.666 -1.668467805  0.336467805 0.6345514
## 5-4    0.612 -0.390467805  1.614467805 0.7687300
## 6-4    0.530 -0.472467805  1.532467805 0.9164375
## 7-4    0.434 -0.568467805  1.436467805 0.9874333
## 8-4    0.238 -0.764467805  1.240467805 0.9999968
## 9-4    0.592 -0.410467805  1.594467805 0.8122238
## 10-4   0.134 -0.868467805  1.136467805 1.0000000
## 11-4  -0.442 -1.444467805  0.560467805 0.9847295
## 12-4  -0.920 -1.922467805  0.082467805 0.1123528
## 13-4  -0.572 -1.574467805  0.430467805 0.8511779
## 14-4  -0.864 -1.866467805  0.138467805 0.1831362
## 15-4  -1.264 -2.266467805 -0.261532195 0.0022166
## 16-4  -1.170 -2.172467805 -0.167532195 0.0073654
## 17-4  -1.312 -2.314467805 -0.309532195 0.0011669
## 18-4  -1.492 -2.494467805 -0.489532195 0.0000916
## 19-4  -0.176 -1.178467805  0.826467805 1.0000000
## 20-4  -0.452 -1.454467805  0.550467805 0.9807230
## 6-5   -0.082 -1.084467805  0.920467805 1.0000000
## 7-5   -0.178 -1.180467805  0.824467805 1.0000000
## 8-5   -0.374 -1.376467805  0.628467805 0.9977809
## 9-5   -0.020 -1.022467805  0.982467805 1.0000000
## 10-5  -0.478 -1.480467805  0.524467805 0.9664464
## 11-5  -1.054 -2.056467805 -0.051532195 0.0287341
## 12-5  -1.532 -2.534467805 -0.529532195 0.0000507
## 13-5  -1.184 -2.186467805 -0.181532195 0.0061905
## 14-5  -1.476 -2.478467805 -0.473532195 0.0001157
## 15-5  -1.876 -2.878467805 -0.873532195 0.0000002
## 16-5  -1.782 -2.784467805 -0.779532195 0.0000011
## 17-5  -1.924 -2.926467805 -0.921532195 0.0000001
## 18-5  -2.104 -3.106467805 -1.101532195 0.0000000
## 19-5  -0.788 -1.790467805  0.214467805 0.3246337
## 20-5  -1.064 -2.066467805 -0.061532195 0.0257056
## 7-6   -0.096 -1.098467805  0.906467805 1.0000000
## 8-6   -0.292 -1.294467805  0.710467805 0.9999244
## 9-6    0.062 -0.940467805  1.064467805 1.0000000
## 10-6  -0.396 -1.398467805  0.606467805 0.9955499
## 11-6  -0.972 -1.974467805  0.030467805 0.0682221
## 12-6  -1.450 -2.452467805 -0.447532195 0.0001687
## 13-6  -1.102 -2.104467805 -0.099532195 0.0166554
## 14-6  -1.394 -2.396467805 -0.391532195 0.0003753
## 15-6  -1.794 -2.796467805 -0.791532195 0.0000009
## 16-6  -1.700 -2.702467805 -0.697532195 0.0000039
## 17-6  -1.842 -2.844467805 -0.839532195 0.0000004
## 18-6  -2.022 -3.024467805 -1.019532195 0.0000000
## 19-6  -0.706 -1.708467805  0.296467805 0.5279545
## 20-6  -0.982 -1.984467805  0.020467805 0.0616959
## 8-7   -0.196 -1.198467805  0.806467805 0.9999999
## 9-7    0.158 -0.844467805  1.160467805 1.0000000
## 10-7  -0.300 -1.302467805  0.702467805 0.9998878
## 11-7  -0.876 -1.878467805  0.126467805 0.1656786
## 12-7  -1.354 -2.356467805 -0.351532195 0.0006564
## 13-7  -1.006 -2.008467805 -0.003532195 0.0481907
## 14-7  -1.298 -2.300467805 -0.295532195 0.0014096
## 15-7  -1.698 -2.700467805 -0.695532195 0.0000041
## 16-7  -1.604 -2.606467805 -0.601532195 0.0000172
## 17-7  -1.746 -2.748467805 -0.743532195 0.0000019
## 18-7  -1.926 -2.928467805 -0.923532195 0.0000001
## 19-7  -0.610 -1.612467805  0.392467805 0.7732656
## 20-7  -0.886 -1.888467805  0.116467805 0.1521161
## 9-8    0.354 -0.648467805  1.356467805 0.9988995
## 10-8  -0.104 -1.106467805  0.898467805 1.0000000
## 11-8  -0.680 -1.682467805  0.322467805 0.5974170
## 12-8  -1.158 -2.160467805 -0.155532195 0.0085354
## 13-8  -0.810 -1.812467805  0.192467805 0.2782223
## 14-8  -1.102 -2.104467805 -0.099532195 0.0166554
## 15-8  -1.502 -2.504467805 -0.499532195 0.0000791
## 16-8  -1.408 -2.410467805 -0.405532195 0.0003078
## 17-8  -1.550 -2.552467805 -0.547532195 0.0000388
## 18-8  -1.730 -2.732467805 -0.727532195 0.0000025
## 19-8  -0.414 -1.416467805  0.588467805 0.9925451
## 20-8  -0.690 -1.692467805  0.312467805 0.5706969
## 10-9  -0.458 -1.460467805  0.544467805 0.9779512
## 11-9  -1.034 -2.036467805 -0.031532195 0.0357707
## 12-9  -1.512 -2.514467805 -0.509532195 0.0000682
## 13-9  -1.164 -2.166467805 -0.161532195 0.0079302
## 14-9  -1.456 -2.458467805 -0.453532195 0.0001547
## 15-9  -1.856 -2.858467805 -0.853532195 0.0000003
## 16-9  -1.762 -2.764467805 -0.759532195 0.0000015
## 17-9  -1.904 -2.906467805 -0.901532195 0.0000002
## 18-9  -2.084 -3.086467805 -1.081532195 0.0000000
## 19-9  -0.768 -1.770467805  0.234467805 0.3703485
## 20-9  -1.044 -2.046467805 -0.041532195 0.0320800
## 11-10 -0.576 -1.578467805  0.426467805 0.8437762
## 12-10 -1.054 -2.056467805 -0.051532195 0.0287341
## 13-10 -0.706 -1.708467805  0.296467805 0.5279545
## 14-10 -0.998 -2.000467805  0.004467805 0.0523734
## 15-10 -1.398 -2.400467805 -0.395532195 0.0003547
## 16-10 -1.304 -2.306467805 -0.301532195 0.0013002
## 17-10 -1.446 -2.448467805 -0.443532195 0.0001788
## 18-10 -1.626 -2.628467805 -0.623532195 0.0000123
## 19-10 -0.310 -1.312467805  0.692467805 0.9998203
## 20-10 -0.586 -1.588467805  0.416467805 0.8244126
## 12-11 -0.478 -1.480467805  0.524467805 0.9664464
## 13-11 -0.130 -1.132467805  0.872467805 1.0000000
## 14-11 -0.422 -1.424467805  0.580467805 0.9907564
## 15-11 -0.822 -1.824467805  0.180467805 0.2547418
## 16-11 -0.728 -1.730467805  0.274467805 0.4699961
## 17-11 -0.870 -1.872467805  0.132467805 0.1742442
## 18-11 -1.050 -2.052467805 -0.047532195 0.0300328
## 19-11  0.266 -0.736467805  1.268467805 0.9999814
## 20-11 -0.010 -1.012467805  0.992467805 1.0000000
## 13-12  0.348 -0.654467805  1.350467805 0.9991206
## 14-12  0.056 -0.946467805  1.058467805 1.0000000
## 15-12 -0.344 -1.346467805  0.658467805 0.9992456
## 16-12 -0.250 -1.252467805  0.752467805 0.9999929
## 17-12 -0.392 -1.394467805  0.610467805 0.9960570
## 18-12 -0.572 -1.574467805  0.430467805 0.8511779
## 19-12  0.744 -0.258467805  1.746467805 0.4290263
## 20-12  0.468 -0.534467805  1.470467805 0.9726574
## 14-13 -0.292 -1.294467805  0.710467805 0.9999244
## 15-13 -0.692 -1.694467805  0.310467805 0.5653466
## 16-13 -0.598 -1.600467805  0.404467805 0.7996250
## 17-13 -0.740 -1.742467805  0.262467805 0.4391487
## 18-13 -0.920 -1.922467805  0.082467805 0.1123528
## 19-13  0.396 -0.606467805  1.398467805 0.9955499
## 20-13  0.120 -0.882467805  1.122467805 1.0000000
## 15-14 -0.400 -1.402467805  0.602467805 0.9949894
## 16-14 -0.306 -1.308467805  0.696467805 0.9998507
## 17-14 -0.448 -1.450467805  0.554467805 0.9824141
## 18-14 -0.628 -1.630467805  0.374467805 0.7311222
## 19-14  0.688 -0.314467805  1.690467805 0.5760465
## 20-14  0.412 -0.590467805  1.414467805 0.9929447
## 16-15  0.094 -0.908467805  1.096467805 1.0000000
## 17-15 -0.048 -1.050467805  0.954467805 1.0000000
## 18-15 -0.228 -1.230467805  0.774467805 0.9999984
## 19-15  1.088  0.085532195  2.090467805 0.0195809
## 20-15  0.812 -0.190467805  1.814467805 0.2742171
## 17-16 -0.142 -1.144467805  0.860467805 1.0000000
## 18-16 -0.322 -1.324467805  0.680467805 0.9996933
## 19-16  0.994 -0.008467805  1.996467805 0.0545811
## 20-16  0.718 -0.284467805  1.720467805 0.4961610
## 18-17 -0.180 -1.182467805  0.822467805 1.0000000
## 19-17  1.136  0.133532195  2.138467805 0.0111426
## 20-17  0.860 -0.142467805  1.862467805 0.1892474
## 19-18  1.316  0.313532195  2.318467805 0.0011053
## 20-18  1.040  0.037532195  2.042467805 0.0335133
## 20-19 -0.276 -1.278467805  0.726467805 0.9999673
tapply(tiempo, grupos, mean)
##     1     2     3     4     5     6     7     8     9    10    11    12    13 
## 5.368 5.016 5.558 5.344 5.956 5.874 5.778 5.582 5.936 5.478 4.902 4.424 4.772 
##    14    15    16    17    18    19    20 
## 4.480 4.080 4.174 4.032 3.852 5.168 4.892
tapply(tiempo, grupos, min)
##    1    2    3    4    5    6    7    8    9   10   11   12   13   14   15   16 
## 5.02 4.52 5.12 5.03 5.37 5.11 5.26 5.15 5.52 4.56 4.28 3.68 4.29 4.14 3.82 3.69 
##   17   18   19   20 
## 3.49 3.42 5.06 4.08
minimos=tapply(tiempo, grupos, min)
min(minimos)
## [1] 3.42
tapply(tiempo, grupos, max)
##    1    2    3    4    5    6    7    8    9   10   11   12   13   14   15   16 
## 5.66 5.51 5.99 6.08 6.61 6.56 6.30 5.85 6.24 6.40 5.38 5.24 5.33 4.73 4.58 4.97 
##   17   18   19   20 
## 4.29 4.67 5.24 5.26
maximos=tapply(tiempo, grupos, max)
max(maximos)
## [1] 6.61

Podemos observar grandes diferencia entrw los valores según al grupo al que pertenezca por lo que sería necesario ver que factores están influenciando en estas diferencias y corregirlas.