Inferencia Estadística

Clase 9: Distribución de la Varianza Muestral

Ing. Eduardo Véliz Quintero, M.Sc.

Varianza Muestral

𝑛 𝑁
1 1
∑ 𝜎 = ∑ ( 𝑋 𝑖 − 𝜇 )2
2 2 2
𝑆= ( 𝑋𝑖− 𝑋 )
𝑛− 1 𝑖=1 𝑁 𝑖=1

E(estimador) = parámetro

Estimador Insesgado
Estimador Insesgado de la Varianza
Distribución -cuadrado
Distribución de la Varianza Muestral
Distribución -cuadrado
Distribución -cuadrado
Intervalos de
Confianza
Intervalo de Confianza de la Varianza
Conf = 0.95
1-conf =0.05
(1-conf)/2 = 0.025
1- (1-conf)/2 =0.975
n=20
S=0.05 intervalo_varianza <- function(S,n,conf){
D1 = qchisq(1-(1-conf)/2,n-1)
Confianza= 95%
D2= qchisq((1-conf)/2,n-1)

inf = ((n-1)*S^2)/D1
sup = ((n-1)*S^2)/D2
c(inf, sup)
}

intervalo_varianza(0.05,20,0.95)

[1] 0.001445864 0.005333174

Ejemplo
Ejemplo
> X=c(47,50,53,45,40,43,39,37
> V=var(X)
> S=sd(X)
>V
[1] 31.07143
>S
[1] 5.574175

> D1 = qchisq(0.975,n-1) > intervalo=c(inf,sup)

> D2 = qchisq(0.025,n-1) > intervalo
> [1] 13.58291 128.70819
> inf = (n-1)*V/D1 > sqrt(intervalo)
> sup = (n-1)*V/D2 [1] 3.685501 11.344963
>
Ejemplo

> X=c(1.40,1.40,1.45,1.49,1.37,1.27,1.40,1.55,1.40,1.42,1.48,1.63) > D1=qchisq(0.975,11)

> V=var(X) > D2=qchisq(0.025,11)
> S=sd(X) >
>V > inf = (11)*V/D1
[1] 0.008451515 > sup = (11)*V/D2
>S
[1] 0.09193212
>
> intervalo=c(inf,sup)
> intervalo
[1] 0.004241171 0.024363941
> sqrt(intervalo)
[1] 0.06512427 0.15608953
Pruebas de
Hipótesis
gl/prob 0,995 0,99 0,975 0,95 0,9 0,1 0,05 0,025 0,01 0,005 gl/prob 0,995 0,99 0,975 0,95 0,9 0,1 0,05 0,025 0,01 0,005
1 7,8794 6,6349 5,0239 3,8415 2,7055 0,0158 0,0039 0,001 0,0002 0 1 0,000 0,000 0,001 0,004 0,016 2,706 3,841 5,024 6,635 7,879
2 10,5966 9,2103 7,3778 5,9915 4,6052 0,2107 0,1026 0,0506 0,0201 0,01 2 0,010 0,020 0,051 0,103 0,211 4,605 5,991 7,378 9,210 10,597
3 12,8382 11,3449 9,3484 7,8147 6,2514 0,5844 0,3518 0,2158 0,1148 0,0717 3 0,072 0,115 0,216 0,352 0,584 6,251 7,815 9,348 11,345 12,838
4 14,8603 13,2767 11,1433 9,4877 7,7794 1,0636 0,7107 0,4844 0,2971 0,207 4 0,207 0,297 0,484 0,711 1,064 7,779 9,488 11,143 13,277 14,860
5 16,7496 15,0863 12,8325 11,0705 9,2364 1,6103 1,1455 0,8312 0,5543 0,4117 5 0,412 0,554 0,831 1,145 1,610 9,236 11,070 12,833 15,086 16,750
6 18,5476 16,8119 14,4494 12,5916 10,6446 2,2041 1,6354 1,2373 0,8721 0,6757 6 0,676 0,872 1,237 1,635 2,204 10,645 12,592 14,449 16,812 18,548
7 20,2777 18,4753 16,0128 14,0671 12,017 2,8331 2,1673 1,6899 1,239 0,9893 7 0,989 1,239 1,690 2,167 2,833 12,017 14,067 16,013 18,475 20,278
8 21,955 20,0902 17,5345 15,5073 13,3616 3,4895 2,7326 2,1797 1,6465 1,3444 8 1,344 1,646 2,180 2,733 3,490 13,362 15,507 17,535 20,090 21,955
9 23,5894 21,666 19,0228 16,919 14,6837 4,1682 3,3251 2,7004 2,0879 1,7349 9 1,735 2,088 2,700 3,325 4,168 14,684 16,919 19,023 21,666 23,589
10 25,1882 23,2093 20,4832 18,307 15,9872 4,8652 3,9403 3,247 2,5582 2,1559 10 2,156 2,558 3,247 3,940 4,865 15,987 18,307 20,483 23,209 25,188
11 26,7568 24,725 21,92 19,6751 17,275 5,5778 4,5748 3,8157 3,0535 2,6032 11 2,603 3,053 3,816 4,575 5,578 17,275 19,675 21,920 24,725 26,757
12 28,2995 26,217 23,3367 21,0261 18,5493 6,3038 5,226 4,4038 3,5706 3,0738 12 3,074 3,571 4,404 5,226 6,304 18,549 21,026 23,337 26,217 28,300
13 29,8195 27,6882 24,7356 22,362 19,8119 7,0415 5,8919 5,0088 4,1069 3,565 13 3,565 4,107 5,009 5,892 7,042 19,812 22,362 24,736 27,688 29,819
14 31,3193 29,1412 26,1189 23,6848 21,0641 7,7895 6,5706 5,6287 4,6604 4,0747 14 4,075 4,660 5,629 6,571 7,790 21,064 23,685 26,119 29,141 31,319
15 32,8013 30,5779 27,4884 24,9958 22,3071 8,5468 7,2609 6,2621 5,2293 4,6009 15 4,601 5,229 6,262 7,261 8,547 22,307 24,996 27,488 30,578 32,801
16 34,2672 31,9999 28,8454 26,2962 23,5418 9,3122 7,9616 6,9077 5,8122 5,1422 16 5,142 5,812 6,908 7,962 9,312 23,542 26,296 28,845 32,000 34,267
17 35,7185 33,4087 30,191 27,5871 24,769 10,0852 8,6718 7,5642 6,4078 5,6972 17 5,697 6,408 7,564 8,672 10,085 24,769 27,587 30,191 33,409 35,718
18 37,1565 34,8053 31,5264 28,8693 25,9894 10,8649 9,3905 8,2307 7,0149 6,2648 18 6,265 7,015 8,231 9,390 10,865 25,989 28,869 31,526 34,805 37,156
19 38,5823 36,1909 32,8523 30,1435 27,2036 11,6509 10,117 8,9065 7,6327 6,844 19 6,844 7,633 8,907 10,117 11,651 27,204 30,144 32,852 36,191 38,582
20 39,9968 37,5662 34,1696 31,4104 28,412 12,4426 10,8508 9,5908 8,2604 7,4338 20 7,434 8,260 9,591 10,851 12,443 28,412 31,410 34,170 37,566 39,997
21 41,4011 38,9322 35,4789 32,6706 29,6151 13,2396 11,5913 10,2829 8,8972 8,0337 21 8,034 8,897 10,283 11,591 13,240 29,615 32,671 35,479 38,932 41,401
22 42,7957 40,2894 36,7807 33,9244 30,8133 14,0415 12,338 10,9823 9,5425 8,6427 22 8,643 9,542 10,982 12,338 14,041 30,813 33,924 36,781 40,289 42,796
23 44,1813 41,6384 38,0756 35,1725 32,0069 14,848 13,0905 11,6886 10,1957 9,2604 23 9,260 10,196 11,689 13,091 14,848 32,007 35,172 38,076 41,638 44,181
24 45,5585 42,9798 39,3641 36,415 33,1962 15,6587 13,8484 12,4012 10,8564 9,8862 24 9,886 10,856 12,401 13,848 15,659 33,196 36,415 39,364 42,980 45,559
25 46,9279 44,3141 40,6465 37,6525 34,3816 16,4734 14,6114 13,1197 11,524 10,5197 25 10,520 11,524 13,120 14,611 16,473 34,382 37,652 40,646 44,314 46,928
26 48,2899 45,6417 41,9232 38,8851 35,5632 17,2919 15,3792 13,8439 12,1981 11,1602 26 11,160 12,198 13,844 15,379 17,292 35,563 38,885 41,923 45,642 48,290
27 49,6449 46,9629 43,1945 40,1133 36,7412 18,1139 16,1514 14,5734 12,8785 11,8076 27 11,808 12,879 14,573 16,151 18,114 36,741 40,113 43,195 46,963 49,645
28 50,9934 48,2782 44,4608 41,3371 37,9159 18,9392 16,9279 15,3079 13,5647 12,4613 28 12,461 13,565 15,308 16,928 18,939 37,916 41,337 44,461 48,278 50,993
29 52,3356 49,5879 45,7223 42,557 39,0875 19,7677 17,7084 16,0471 14,2565 13,1211 29 13,121 14,256 16,047 17,708 19,768 39,087 42,557 45,722 49,588 52,336
30 53,672 50,8922 46,9792 43,773 40,256 20,5992 18,4927 16,7908 14,9535 13,7867 30 13,787 14,953 16,791 18,493 20,599 40,256 43,773 46,979 50,892 53,672
Pruebas de Hipótesis
prueba_hipotesis_varianza <-function(S,n,a_var,conf,colas){
ep=((n-1)*S^2)/a_var

if (colas == 2){
K1= qchisq((1-conf)/2,n-1)
K2 = qchisq(1-(1-conf)/2,n-1)
} else {
K1= qchisq((1-conf),n-1)
K2 = qchisq(conf,n-1)
}

if (colas==2){
if(ep>=K2) vp=1-pchisq(ep,n-1) else vp=pchisq(ep,n-1)
if (ep<=K1 || ep>=K2) print("Rechazar H0") else print("Aceptar H0")
} else if (colas==-1){
vp=pchisq(ep,n-1)
if (ep<=K1) print("Rechazar H0") else print("Aceptar H0")
} else if (colas==1){
vp=1-pchisq(ep,n-1) prueba_hipotesis_varianza(0.05,20,1,0.95,2)
if (ep>=K2) print("Rechazar H0") else print("Aceptar H0")
}

print(c(K1,ep,K2))
vp
}
Ejemplo

24.99579

7.260944
Ejemplo

17.70837
Ejemplo

S2=0.8106061

9.485816

3.815748 21.92005
¿Dudas?