1.Estimación del Modelo.

# Estimar el modelo: price en función de lotsize, sqrft y bdrms
data(hprice1)
modelo_hprice <- lm(price ~ lotsize + sqrft + bdrms, data = hprice1)
# Presentación de resultados estilo stargazer
stargazer(modelo_hprice, title = "Modelo para hprice1", type = "html")
Modelo para hprice1
Dependent variable:
price
lotsize 0.002***
(0.001)
sqrft 0.123***
(0.013)
bdrms 13.853
(9.010)
Constant -21.770
(29.475)
Observations 88
R2 0.672
Adjusted R2 0.661
Residual Std. Error 59.833 (df = 84)
F Statistic 57.460*** (df = 3; 84)
Note: p<0.1; p<0.05; p<0.01 
# Extracción de los residuos para las pruebas de normalidad
residuos <- modelo_hprice$residuals

2. Verficación del Supuesto de Normalidad

# Verificando el ajuste con fitdistrplus
fit_normal <- fitdist(data = residuos, distr = "norm")
plot(fit_normal)

summary(fit_normal)
## Fitting of the distribution ' norm ' by maximum likelihood 
## Parameters : 
##           estimate Std. Error
## mean -2.321494e-15   6.231625
## sd    5.845781e+01   4.406424
## Loglikelihood:  -482.8775   AIC:  969.7549   BIC:  974.7096 
## Correlation matrix:
##      mean sd
## mean    1  0
## sd      0  1

a) La prueba Jarque-Bera (JB)

# Prueba JB usando tseries
options(scipen = 999)
salida_JB <- jarque.bera.test(residuos)
salida_JB
## 
##  Jarque Bera Test
## 
## data:  residuos
## X-squared = 32.278, df = 2, p-value = 0.00000009794
# Gráfica con fastGraph
options(scipen = 999)
alpha_sig <- 0.05
JB <- salida_JB$statistic
gl <- salida_JB$parameter
VC <- qchisq(1 - alpha_sig, gl, lower.tail = TRUE)

shadeDist(JB, ddist = "dchisq", parm1 = gl, lower.tail = FALSE, xmin = 0,
          sub = paste("VC:", round(VC, 2), " ", "JB:", round(JB, 2)))

# Resumen manual de los residuos para JB usando dplyr
residuos %>%
  as_tibble() %>%
  summarise(n = n(),
            Media = mean(value),
            Desv.Est = sd(value)) %>%
  gt() %>% 
  tab_header("Resumen Manual para JB")
Resumen Manual para JB
n Media Desv.Est
88 -0.000000000000002321494 58.79282

Conclusión: Dado que el p-value 0.00000009794 es menor al nivel de significancia 0.05, se rechaza la hipótesis nula . Por lo tanto, bajo la prueba Jarque-Bera, los residuos no siguen una distribución normal, lo cual se confirma visualmente en la gráfica de Chi-cuadrado donde el estadístico JB cae muy lejos del valor crítico.

b) La prueba KS

# Tabla de cálculo manual KS usando dplyr y gt
tabla_KS <- residuos %>%
  as_tibble() %>%
  mutate(posicion = row_number()) %>%
  arrange(value) %>%
  mutate(dist1 = row_number() / n(),
         dist2 = (row_number() - 1) / n(),
         zi = as.vector(scale(value, center = TRUE)),
         pi = pnorm(zi, lower.tail = TRUE),
         dif1 = abs(dist1 - pi),
         dif2 = abs(dist2 - pi)) %>%
  rename(residuales = value)

# Aplicación de formato gt y resaltado de máximos
tabla_KS %>%
  gt() %>%
  tab_header("Tabla para calcular el Estadistico KS") %>%
  tab_source_note(source_note = "Fuente: Elaboración propia") %>%
  tab_style(
    style = list(cell_fill(color = "#A569BD"), cell_text(style = "italic")),
    locations = cells_body(columns = dif1, rows = dif1 == max(dif1))
  ) %>%
  tab_style(
    style = list(cell_fill(color = "#3498DB"), cell_text(style = "italic")),
    locations = cells_body(columns = dif2, rows = dif2 == max(dif2))
  )
Tabla para calcular el Estadistico KS
residuales posicion dist1 dist2 zi pi dif1 dif2
-120.026447 81 0.01136364 0.00000000 -2.041515459 0.02059981 0.0092361731 0.0205998094
-115.508697 77 0.02272727 0.01136364 -1.964673586 0.02472601 0.0019987418 0.0133623781
-107.080889 24 0.03409091 0.02272727 -1.821326006 0.03427866 0.0001877487 0.0115513850
-91.243980 48 0.04545455 0.03409091 -1.551957925 0.06033615 0.0148816002 0.0262452366
-85.461169 12 0.05681818 0.04545455 -1.453598781 0.07302879 0.0162106057 0.0275742421
-77.172687 32 0.06818182 0.05681818 -1.312620980 0.09465535 0.0264735301 0.0378371665
-74.702719 54 0.07954545 0.06818182 -1.270609602 0.10193378 0.0223883300 0.0337519664
-65.502849 39 0.09090909 0.07954545 -1.114130117 0.13261169 0.0417025941 0.0530662305
-63.699108 69 0.10227273 0.09090909 -1.083450505 0.13930425 0.0370315271 0.0483951634
-62.566594 83 0.11363636 0.10227273 -1.064187703 0.14362184 0.0299854747 0.0413491110
-59.845223 36 0.12500000 0.11363636 -1.017900230 0.15436269 0.0293626861 0.0407263225
-54.466158 13 0.13636364 0.12500000 -0.926408352 0.17711690 0.0407532663 0.0521169027
-54.300415 14 0.14772727 0.13636364 -0.923589260 0.17785010 0.0301228311 0.0414864675
-52.129801 15 0.15909091 0.14772727 -0.886669532 0.18762842 0.0285375141 0.0399011505
-51.441108 17 0.17045455 0.15909091 -0.874955638 0.19079902 0.0203444766 0.0317081129
-48.704980 47 0.18181818 0.17045455 -0.828417174 0.20371714 0.0218989601 0.0332625965
-48.350295 29 0.19318182 0.18181818 -0.822384375 0.20542908 0.0122472664 0.0236109028
-47.855859 11 0.20454545 0.19318182 -0.813974573 0.20782976 0.0032843043 0.0146479407
-45.639765 1 0.21590909 0.20454545 -0.776281294 0.21879146 0.0028823668 0.0142460032
-43.142550 9 0.22727273 0.21590909 -0.733806463 0.23153335 0.0042606233 0.0156242596
-41.749618 57 0.23863636 0.22727273 -0.710114247 0.23881665 0.0001802823 0.0115439187
-40.869022 27 0.25000000 0.23863636 -0.695136302 0.24348494 0.0065150566 0.0048485798
-37.749811 34 0.26136364 0.25000000 -0.642082009 0.26040997 0.0009536682 0.0104099682
-36.663785 71 0.27272727 0.26136364 -0.623609925 0.26644190 0.0062853771 0.0050782592
-36.646568 79 0.28409091 0.27272727 -0.623317083 0.26653809 0.0175528221 0.0061891857
-33.801248 37 0.29545455 0.28409091 -0.574921384 0.28267223 0.0127823120 0.0014186757
-29.766931 16 0.30681818 0.29545455 -0.506302171 0.30632227 0.0004959124 0.0108677240
-26.696234 22 0.31818182 0.30681818 -0.454073044 0.32488813 0.0067063089 0.0180699452
-24.271531 23 0.32954545 0.31818182 -0.412831567 0.33986501 0.0103195566 0.0216831929
-23.651448 86 0.34090909 0.32954545 -0.402284648 0.34373728 0.0028281851 0.0141918214
-19.683427 88 0.35227273 0.34090909 -0.334793052 0.36889060 0.0166178738 0.0279815102
-17.817835 10 0.36363636 0.35227273 -0.303061413 0.38092153 0.0172851663 0.0286488027
-16.762094 60 0.37500000 0.36363636 -0.285104441 0.38778206 0.0127820638 0.0241457002
-16.596960 21 0.38636364 0.37500000 -0.282295711 0.38885839 0.0024947507 0.0138583870
-16.271207 58 0.39772727 0.38636364 -0.276755010 0.39098411 0.0067431583 0.0046204781
-13.815798 56 0.40909091 0.39772727 -0.234991254 0.40710776 0.0019831485 0.0093804879
-13.462160 75 0.42045455 0.40909091 -0.228976273 0.40944368 0.0110108666 0.0003527698
-12.081520 4 0.43181818 0.42045455 -0.205493119 0.41859344 0.0132247451 0.0018611087
-11.629207 51 0.44318182 0.43181818 -0.197799788 0.42160086 0.0215809622 0.0102173258
-11.312669 74 0.45454545 0.44318182 -0.192415834 0.42370825 0.0308372092 0.0194735728
-8.236558 3 0.46590909 0.45454545 -0.140094626 0.44429261 0.0216164775 0.0102528411
-7.662789 70 0.47727273 0.46590909 -0.130335452 0.44815052 0.0291222111 0.0177585748
-6.752801 67 0.48863636 0.47727273 -0.114857588 0.45427900 0.0343573625 0.0229937262
-6.707262 31 0.50000000 0.48863636 -0.114083016 0.45458599 0.0454140074 0.0340503710
-6.402439 85 0.51136364 0.50000000 -0.108898313 0.45664157 0.0547220642 0.0433584278
-5.446904 82 0.52272727 0.51136364 -0.092645733 0.46309251 0.0596347676 0.0482711313
-3.537785 43 0.53409091 0.52272727 -0.060173762 0.47600862 0.0580822876 0.0467186512
-2.824941 61 0.54545455 0.53409091 -0.048049090 0.48083856 0.0646159857 0.0532523493
-2.745208 68 0.55681818 0.54545455 -0.046692922 0.48137899 0.0754391961 0.0640755598
-0.195089 65 0.56818182 0.55681818 -0.003318245 0.49867621 0.0695056040 0.0581419676
1.399296 55 0.57954545 0.56818182 0.023800450 0.50949411 0.0700513452 0.0586877088
5.363331 26 0.59090909 0.57954545 0.091224254 0.53634280 0.0545662924 0.0432026561
6.700640 53 0.60227273 0.59090909 0.113970383 0.54536936 0.0569033628 0.0455397265
7.386314 80 0.61363636 0.60227273 0.125632935 0.54998875 0.0636476093 0.0522839730
9.099900 41 0.62500000 0.61363636 0.154779103 0.56150227 0.0634977329 0.0521340965
12.433611 46 0.63636364 0.62500000 0.211481796 0.58374433 0.0526193043 0.0412556680
16.718018 62 0.64772727 0.63636364 0.284354766 0.61193074 0.0357965328 0.0244328965
18.093192 5 0.65909091 0.64772727 0.307744934 0.62086179 0.0382291219 0.0268654856
18.801816 38 0.67045455 0.65909091 0.319797835 0.62543921 0.0450153400 0.0336517036
19.168108 33 0.68181818 0.67045455 0.326028052 0.62779843 0.0540197476 0.0426561112
19.219211 72 0.69318182 0.68181818 0.326897255 0.62812720 0.0650546167 0.0536909803
20.334434 59 0.70454545 0.69318182 0.345865960 0.63527827 0.0692671805 0.0579035442
24.909926 78 0.71590909 0.70454545 0.423689939 0.66410402 0.0518050676 0.0404414312
26.236229 40 0.72727273 0.71590909 0.446248874 0.67229126 0.0549814685 0.0436178321
30.924022 25 0.73863636 0.72727273 0.525982978 0.70054998 0.0380863808 0.0267227444
32.253952 45 0.75000000 0.73863636 0.548603608 0.70836125 0.0416387548 0.0302751184
32.529367 49 0.76136364 0.75000000 0.553288104 0.70996693 0.0513967091 0.0400330727
32.675968 18 0.77272727 0.76136364 0.555781630 0.71081993 0.0619073452 0.0505437088
33.275839 20 0.78409091 0.77272727 0.565984762 0.71429793 0.0697929786 0.0584293423
36.031430 52 0.79545455 0.78409091 0.612854281 0.73001365 0.0654408934 0.0540772571
37.147186 84 0.80681818 0.79545455 0.631832029 0.73625168 0.0705665028 0.0592028664
40.320875 7 0.81818182 0.80681818 0.685812928 0.75358446 0.0645973596 0.0532337232
44.334467 30 0.82954545 0.81818182 0.754079634 0.77459930 0.0549461574 0.0435825211
46.907165 28 0.84090909 0.82954545 0.797838357 0.78751785 0.0533912405 0.0420276041
54.418366 87 0.85227273 0.84090909 0.925595465 0.82267187 0.0296008528 0.0182372164
55.091131 35 0.86363636 0.85227273 0.937038450 0.82563061 0.0380057535 0.0266421172
55.470305 44 0.87500000 0.86363636 0.943487765 0.82728426 0.0477157353 0.0363520989
62.939597 6 0.88636364 0.87500000 1.070532059 0.85781006 0.0285535797 0.0171899433
66.478628 50 0.89772727 0.88636364 1.130727018 0.87091500 0.0268122757 0.0154486394
67.426518 63 0.90909091 0.89772727 1.146849569 0.87427810 0.0348128083 0.0234491719
67.603959 19 0.92045455 0.90909091 1.149867648 0.87490081 0.0455537393 0.0341901029
69.707122 64 0.93181818 0.92045455 1.185640095 0.88211777 0.0497004123 0.0383367759
69.843246 8 0.94318182 0.93181818 1.187955411 0.88257451 0.0606073068 0.0492436705
74.848732 2 0.95454545 0.94318182 1.273093116 0.89850750 0.0560379553 0.0446743189
112.729191 66 0.96590909 0.95454545 1.917397313 0.97240626 0.0064971714 0.0178608078
163.795081 73 0.97727273 0.96590909 2.785970904 0.99733162 0.0200588896 0.0314225260
198.660139 42 0.98863636 0.97727273 3.378986513 0.99963623 0.0109998685 0.0223635048
209.375830 76 1.00000000 0.98863636 3.561248407 0.99981545 0.0001845478 0.0111790885
Fuente: Elaboración propia 
# Prueba KS-Lilliefors automática
prueba_KS <- lillie.test(residuos)
prueba_KS
## 
##  Lilliefors (Kolmogorov-Smirnov) normality test
## 
## data:  residuos
## D = 0.075439, p-value = 0.2496

Conclusión: En este caso, el estadístico D obtenido es 0.0754, el cual es menor al Valor Crítico calculado de 0.0944. Asimismo, el p-value resultante es 0.2496, el cual es mayor al nivel de significancia de 0.05. Por lo tanto, no se rechaza la Hipótesis Nula y se concluye que, bajo la prueba de Lilliefors, los residuos siguen una distribución normal.

c) La prueba Shapiro-Wilk (SW)

library(dplyr)
library(gt)
# Cálculo Manual
# 1. Preparación de la tabla base
residuos <- modelo_hprice$residuals
residuos %>%    
  as_tibble() %>%  
  rename(residuales = value) %>%  
  arrange(residuales) %>%  
  mutate(pi = (row_number() - 0.375) / (n() + 0.25)) %>%  
  mutate(mi = qnorm(pi, lower.tail = TRUE)) %>%   
  mutate(ai = 0) -> tabla_SW

# 2.Definición de parámetros para el cálculo de ai
m <- sum(tabla_SW$mi^2)
n <- nrow(hprice1) # Tamaño de muestra de hprice1
theta <- 1 / sqrt(n)

# 3.Cálculo de los coeficientes ai (Fórmulas de aproximación de Royston)
tabla_SW$ai[n] <- -2.706056*theta^5 + 4.434685*theta^4 - 2.071190*theta^3 - 0.147981*theta^2 + 0.2211570*theta + tabla_SW$mi[n]/sqrt(m)
tabla_SW$ai[n-1] <- -3.582633*theta^5 + 5.682633*theta^4 - 1.752461*theta^3 - 0.293762*theta^2 + 0.042981*theta + tabla_SW$mi[n-1]/sqrt(m)
tabla_SW$ai[1] <- -tabla_SW$ai[n]
tabla_SW$ai[2] <- -tabla_SW$ai[n-1]

# Cálculo de ai para los valores intermedios
omega <- (m - 2*tabla_SW$mi[n]^2 - 2*tabla_SW$mi[n-1]^2) / (1 - 2*tabla_SW$ai[n]^2 - 2*tabla_SW$ai[n-1]^2)
tabla_SW$ai[3:(n-2)] <- tabla_SW$mi[3:(n-2)] / sqrt(omega)

# Cálculo de productos y cuadrados para el estadístico W
tabla_SW %>%   
  mutate(ai_ui = ai * residuales, ui2 = residuales^2) -> tabla_SW

# 4. Generación de la Tabla con formato gt
tabla_SW %>%  
  gt() %>% 
  tab_header("Tabla para calcular el Estadistico W") %>%  
  tab_source_note(source_note = "Fuente: Elaboración propia")
Tabla para calcular el Estadistico W
residuales pi mi ai ai_ui ui2
-120.026447 0.007082153 -2.45306927 -0.286093929 34.338837782 14406.34799223
-115.508697 0.018413598 -2.08767462 -0.226331231 26.143225495 13342.25903657
-107.080889 0.029745042 -1.88455395 -0.201511408 21.578020632 11466.31670225
-91.243980 0.041076487 -1.73832835 -0.185875811 16.960048752 8325.46388922
-85.461169 0.052407932 -1.62194155 -0.173430814 14.821600075 7303.61136157
-77.172687 0.063739377 -1.52411994 -0.162970954 12.576906330 5955.62354189
-74.702719 0.075070822 -1.43903134 -0.153872609 11.494702279 5580.49626206
-65.502849 0.086402266 -1.36324747 -0.145769197 9.548297773 4290.62326804
-63.699108 0.097733711 -1.29457343 -0.138426027 8.817614500 4057.57641853
-62.566594 0.109065156 -1.23151500 -0.131683320 8.238976839 3914.57869135
-59.845223 0.120396601 -1.17300649 -0.125427129 7.506214499 3581.45072682
-54.466158 0.131728045 -1.11825971 -0.119573169 6.512691096 2966.56233834
-54.300415 0.143059490 -1.06667420 -0.114057239 6.193355472 2948.53511008
-52.129801 0.154390935 -1.01778137 -0.108829231 5.673246083 2717.51610406
-51.441108 0.165722380 -0.97120790 -0.103849228 5.342119306 2646.18755812
-48.704980 0.177053824 -0.92665123 -0.099084876 4.825926905 2372.17509746
-48.350295 0.188385269 -0.88386232 -0.094509548 4.569564512 2337.75102457
-47.855859 0.199716714 -0.84263354 -0.090101040 4.311862673 2290.18324033
-45.639765 0.211048159 -0.80278966 -0.085840618 3.917745629 2082.98814155
-43.142550 0.222379603 -0.76418130 -0.081712307 3.525277277 1861.27961161
-41.749618 0.233711048 -0.72667986 -0.077702356 3.244043648 1743.03058469
-40.869022 0.245042493 -0.69017366 -0.073798824 3.016085791 1670.27697055
-37.749811 0.256373938 -0.65456498 -0.069991263 2.642156946 1425.04821452
-36.663785 0.267705382 -0.61976766 -0.066270458 2.429725818 1344.23312095
-36.646568 0.279036827 -0.58570518 -0.062628228 2.295109622 1342.97093753
-33.801248 0.290368272 -0.55230918 -0.059057264 1.996209250 1142.52439130
-29.766931 0.301699717 -0.51951819 -0.055550992 1.653582575 886.07020942
-26.696234 0.313031161 -0.48727661 -0.052103467 1.390966354 712.68890388
-24.271531 0.324362606 -0.45553386 -0.048709282 1.182248861 589.10722688
-23.651448 0.335694051 -0.42424369 -0.045363489 1.072912217 559.39099788
-19.683427 0.347025496 -0.39336354 -0.042061540 0.827915257 387.43729851
-17.817835 0.358356941 -0.36285409 -0.038799229 0.691318234 317.47522771
-16.762094 0.369688385 -0.33267878 -0.035572645 0.596272007 280.96778010
-16.596960 0.381019830 -0.30280344 -0.032378138 0.537378676 275.45909399
-16.271207 0.392351275 -0.27319601 -0.029212277 0.475319006 264.75217651
-13.815798 0.403682720 -0.24382619 -0.026071824 0.360203050 190.87627634
-13.462160 0.415014164 -0.21466524 -0.022953704 0.309006447 181.22976154
-12.081520 0.426345609 -0.18568573 -0.019854987 0.239878409 145.96311543
-11.629207 0.437677054 -0.15686137 -0.016772858 0.195055032 135.23845458
-11.312669 0.449008499 -0.12816677 -0.013704604 0.155035654 127.97648221
-8.236558 0.460339943 -0.09957734 -0.010647596 0.087699542 67.84088513
-7.662789 0.471671388 -0.07106908 -0.007599268 0.058231584 58.71832836
-6.752801 0.483002833 -0.04261848 -0.004557105 0.030773222 45.60032533
-6.707262 0.494334278 -0.01420234 -0.001518626 0.010185824 44.98736398
-6.402439 0.505665722 0.01420234 0.001518626 -0.009722911 40.99122172
-5.446904 0.516997167 0.04261848 0.004557105 -0.024822110 29.66876028
-3.537785 0.528328612 0.07106908 0.007599268 -0.026884576 12.51592288
-2.824941 0.539660057 0.09957734 0.010647596 -0.030078835 7.98029397
-2.745208 0.550991501 0.12816677 0.013704604 -0.037621996 7.53616965
-0.195089 0.562322946 0.15686137 0.016772858 -0.003272200 0.03805971
1.399296 0.573654391 0.18568573 0.019854987 0.027782994 1.95802794
5.363331 0.584985836 0.21466524 0.022953704 0.123108313 28.76531940
6.700640 0.596317280 0.24382619 0.026071824 0.174697904 44.89857663
7.386314 0.607648725 0.27319601 0.029212277 0.215771059 54.55763860
9.099900 0.618980170 0.30280344 0.032378138 0.294637808 82.80817401
12.433611 0.630311615 0.33267878 0.035572645 0.442296424 154.59467612
16.718018 0.641643059 0.36285409 0.038799229 0.648646203 279.49212715
18.093192 0.652974504 0.39336354 0.042061540 0.761027520 327.36359375
18.801816 0.664305949 0.42424369 0.045363489 0.852915978 353.50828232
19.168108 0.675637394 0.45553386 0.048709282 0.933664777 367.41636183
19.219211 0.686968839 0.48727661 0.052103467 1.001387528 369.37806665
20.334434 0.698300283 0.51951819 0.055550992 1.129598008 413.48922446
24.909926 0.709631728 0.55230918 0.059057264 1.471112049 620.50439009
26.236229 0.720963173 0.58570518 0.062628228 1.643128534 688.33970624
30.924022 0.732294618 0.61976766 0.066270458 2.049349072 956.29510728
32.253952 0.743626062 0.65456498 0.069991263 2.257494854 1040.31742689
32.529367 0.754957507 0.69017366 0.073798824 2.400629035 1058.15970869
32.675968 0.766288952 0.72667986 0.077702356 2.538999708 1067.71890359
33.275839 0.777620397 0.76418130 0.081712307 2.719045583 1107.28147309
36.031430 0.788951841 0.80278966 0.085840618 3.092960242 1298.26396526
37.147186 0.800283286 0.84263354 0.090101040 3.347000059 1379.91339592
40.320875 0.811614731 0.88386232 0.094509548 3.810707636 1625.77293960
44.334467 0.822946176 0.92665123 0.099084876 4.392875123 1965.54494196
46.907165 0.834277620 0.97120790 0.103849228 4.871272904 2200.28216686
54.418366 0.845609065 1.01778137 0.108829231 5.922308882 2961.35853839
55.091131 0.856940510 1.06667420 0.114057239 6.283542333 3035.03273452
55.470305 0.868271955 1.11825971 0.119573169 6.632760113 3076.95468678
62.939597 0.879603399 1.17300649 0.125427129 7.894332885 3961.39282116
66.478628 0.890934844 1.23151500 0.131683320 8.754126443 4419.40796540
67.426518 0.902266289 1.29457343 0.138426027 9.333585010 4546.33534619
67.603959 0.913597734 1.36324747 0.145769197 9.854574914 4570.29533539
69.707122 0.924929178 1.43903134 0.153872609 10.726016772 4859.08292257
69.843246 0.936260623 1.52411994 0.162970954 11.382420482 4878.07906512
74.848732 0.947592068 1.62194155 0.173430814 12.981076532 5602.33268291
112.729191 0.958923513 1.73832835 0.185875811 20.953629849 12707.87061041
163.795081 0.970254958 1.88455395 0.201511408 33.006577315 26828.82842547
198.660139 0.981586402 2.08767462 0.226331231 44.962993843 39465.85101402
209.375830 0.992917847 2.45306927 0.286093929 59.901153719 43838.23810785
Fuente: Elaboración propia

Cálculo del Estadístico W

# Calculamos el estadístico W manual usando las sumatorias de la tabla
W <- (sum(tabla_SW$ai_ui)^2) / sum(tabla_SW$ui2)
print(W)
## [1] 0.9413208

Cálculo del \(W_n\) y su p-value

# Definimos mu y sigma para la normalización (Aproximación de Royston)
mu <- 0.0038915*log(n)^3 - 0.083751*log(n)^2 - 0.31082*log(n) - 1.5861
sigma <- exp(0.0030302*log(n)^2 - 0.082676*log(n) - 0.4803)

# Calculamos el estadístico normalizado Wn
Wn <- (log(1 - W) - mu) / sigma
print(Wn)
## [1] 3.241867
# Obtenemos el p-value usando la distribución normal
p.value <- pnorm(Wn, lower.tail = FALSE)
print(p.value)
## [1] 0.0005937472

#Gráfica de la prueba Shapiro-Wilk

# Graficamos la distribución normal y sombreamos el área del p-value
library(fastGraph)
shadeDist(Wn, ddist = "dnorm", lower.tail = FALSE)

# Validación automática de Shapiro-Wilk

shapiro.test(residuos)
## 
##  Shapiro-Wilk normality test
## 
## data:  residuos
## W = 0.94132, p-value = 0.0005937

Conclusión: Tras realizar el cálculo manual y validarlo con la librería, se obtuvo un estadístico Wde aproximadamente 0.9412 y un p-value de 0.00059 . Dado que el p-value es menor al nivel de significancia 0.05, se rechaza la Hipótesis Nula . Esto indica que los residuos no siguen una distribución normal. La gráfica muestra que el valor normalizado Wn se encuentra significativamente alejado del centro de la campana, cayendo en la zona de rechazo.