x = c(11.22, 9.56,
      10.06, 10.21, 10.95, 10.03, 10.75, 10.71, 11.42, 9.61, 10.91, 8.14,
      8.95, 10.57, 11.1)
mean= mean(x)
mean

x1 = c(mean(x) -(qnorm((1+0.9973)/2))/sqrt(length(x)), 
       mean(x) + (qnorm((1+0.9973)/2))/sqrt(length(x)))
#x1-- 99% CI

x2=c(mean(x) -(qnorm((1+0.95)/2))/sqrt(length(x)), 
     mean(x) + (qnorm((1+0.95)/2))/sqrt(length(x)))

#x2-- 95% CI

x3=c(mean(x) -(qnorm((1+0.9)/2))/sqrt(length(x)), 
     mean(x) + (qnorm((1+0.9)/2))/sqrt(length(x)))

#x3-- 90% CI

x4=c(mean(x) -(qnorm((1+0.85)/2))/sqrt(length(x)),
     mean(x) + (qnorm((1+0.85)/2))/sqrt(length(x)))

#x4-- 85% CI

x5=c(mean(x) -(qnorm((1+0.8)/2))/sqrt(length(x)),
     mean(x) + (qnorm((1+0.8)/2))/sqrt(length(x)))

cidata= t(cbind(x1,x2,x3, x4,x5))

require(ggplot2)
df = data.frame(x= c(99.73, 95, 90, 85,80), 
                z1 = cidata[,1], z2=cidata[,2], mean )
ggplot(df, aes(x=x, y=mean)) +
  geom_point(size=4, color="#440154") + 
  geom_errorbar(aes(ymax= z2, ymin=z1), color="#3b528b")

-


alpha <- 0.95
c(0, qgamma(alpha, shape = 15))
c(qgamma(1 - alpha, shape = 15), Inf)

beta <- 1 - alpha
c(qgamma(beta/ 2, shape = 15), qgamma(1 - beta / 2, 
                                        shape = 15))


qt(0.975, df = 14)


x = c(11.22, 9.56,
      10.06, 10.21, 10.95, 10.03, 10.75, 10.71, 11.42, 9.61, 10.91, 8.14,
      8.95, 10.57, 11.1)
mean= mean(x)
sd = sd(x)
x1 = c(mean(x) -(qt((1+0.9973)/2,14)*sd)/sqrt(length(x)), 
       mean(x) + (qt((1+0.9973)/2,14)*sd)/sqrt(length(x)))
x2=c(mean(x) -(qt((1+0.95)/2,14)*sd)/sqrt(length(x)), 
     mean(x) + (qt((1+0.95)/2,14)*sd)/sqrt(length(x)))
x3=c(mean(x) -(qt((1+0.9)/2,14)*sd)/sqrt(length(x)), 
     mean(x) + (qt((1+0.9)/2,14)*sd)/sqrt(length(x)))
x4=c(mean(x) -(qt((1+0.85)/2,14)*sd)/sqrt(length(x)), 
     mean(x) + (qt((1+0.85)/2,14)*sd)/sqrt(length(x)))
x5=c(mean(x) -(qt((1+0.8)/2,14)*sd)/sqrt(length(x)), 
     mean(x) + (qt((1+0.8)/2,14)*sd)/sqrt(length(x)))
cidata= t(cbind(x1,x2,x3, x4,x5))


require(ggplot2)
df = data.frame(x= c(99.73, 95, 90, 85,80), 
                z1 = cidata[,1], z2=cidata[,2], mean )
ggplot(df, aes(x=x, y=mean)) + 
  geom_point(size=4, color="#440154") + 
  geom_errorbar(aes(ymax= z2, ymin=z1), color="#3b528b")

cifn = function(x, alpha=0.95){ 
  z = qnorm( (1-alpha)/2, lower.tail=FALSE)
  sdx = sqrt(1/length(x))
  c(mean(x) - z*sdx, mean(x) + z*sdx)
}

x1 = rnorm(100,0,1);y = cifn(x1)
y
x2 = rnorm(100,0,1);z = cifn(x2)
z
x3 = rnorm(100,0,1);w = cifn(x3)
w


par(mfrow=c(1,3))
plot(c(0,0), y, type="l", xlab="", ylab="",  axes=FALSE)
axis(2, at=c(-0.25, -0.15,-0.1,0,0.1,0.15, 0.25) )
points(0,0, pch=19, col="blue")
points(0,y[1], pch=23, col="brown")
points(0,y[2], pch=24, col="brown")

plot(c(0,0), z, type="l", xlab="", ylab="",  axes=FALSE)
axis(2, at=c(-0.25, -0.15,-0.1,0,0.1,0.15, 0.25) )
points(0,0, pch=19, col="blue")
points(0,z[1], pch=23, col="brown")
points(0,z[2], pch=24, col="brown")

plot(c(0,0), w, type="l", xlab="", ylab="",  axes=FALSE)
axis(2, at=c(-0.25, -0.15,-0.1,0,0.1,0.15, 0.25) )
points(0,0, pch=19, col="blue")
points(0,w[1], pch=23, col="brown")
points(0,w[2], pch=24, col="brown")









#computes (MNormal) confidence interval at 95% 
cifn = function(x, alpha=0.95){ 
  z = qnorm( (1-alpha)/2, lower.tail=FALSE)
  sdx = sqrt(1/length(x))
  c(mean(x) - z*sdx, mean(x) + z*sdx)
}


normaldata = replicate(40, rnorm(100,0,1),
                       simplify=FALSE)

cidata = sapply(normaldata, cifn)

TRUEIN = cidata[1,]*cidata[2,]<0
table(TRUEIN)

df = data.frame(x= 1:40, z1 = cidata[1,], z2=cidata[2,])
ggplot(df, aes(x=x, y=0)) + 
  geom_point(size=4, color="#440154") + 
  geom_errorbar(aes(ymax= z2, ymin=z1), color="#3b528b")