Intervalos Proporción

?prop.test

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x<-10
n<-50
prop.test(x,n,p=0.5)

## 
##  1-sample proportions test with continuity correction
## 
## data:  x out of n, null probability 0.5
## X-squared = 16.82, df = 1, p-value = 4.11e-05
## alternative hypothesis: true p is not equal to 0.5
## 95 percent confidence interval:
##  0.1050216 0.3414368
## sample estimates:
##   p 
## 0.2

qbeta(0.025,x+1/2,n-x+1/2)

## [1] 0.107734

qbeta(0.975,x+1/2,n-x+1/2)

## [1] 0.3258269

y<-1
N<-180
prop.test(y,N)

## 
##  1-sample proportions test with continuity correction
## 
## data:  y out of N, null probability 0.5
## X-squared = 174.05, df = 1, p-value < 2.2e-16
## alternative hypothesis: true p is not equal to 0.5
## 95 percent confidence interval:
##  0.000290069 0.035287041
## sample estimates:
##           p 
## 0.005555556

qbeta(0.025,y+1/2,N-y+1/2)

## [1] 0.0006000859

qbeta(0.975,y+1/2,N-y+1/2)

## [1] 0.02566883

# Método de montecarlo
x<-runif(10000)
y<-runif(10000)
plot(x,y)

pi/4

## [1] 0.7853982

sum((x-0.5)**2+(y-0.5)**2 <0.25)

## [1] 7892

7891/10000

## [1] 0.7891

x<-51
y<-49
n<-100
m<-100
p1<-rbeta(10000,x+0.5,n-x+1/2)
p2<-rbeta(10000,y+0.5,m-y+1/2)
quantile(p1-p2,0.025)

##       2.5% 
## -0.1163147

quantile(p1-p2,0.975)

##     97.5% 
## 0.1572224

p1<-0.51
p2<-0.49
p1-p2-1.96*sqrt(p1*(1-p1)/100+p2*(1-p2)/100)

## [1] -0.1185652

p1-p2+1.96*sqrt(p1*(1-p1)/100+p2*(1-p2)/100)

## [1] 0.1585652

#Ejercicio

#1. 0.2,n=2000, 0.6, n=100