Q1

vec1 <- rnorm(1000, 5.7, 0.2)
avg1 <- mean(vec1) # mu (population mean)
plot(density(vec1))
abline(v = avg1, col = "red")

avg1
## [1] 5.693162

Q2

vec2 <- sample(vec1, 50, replace = FALSE) # random subset of vec1
avg2 <- mean(vec2) # xbar (sample mean)
avg2
## [1] 5.685813
ci2 <- t.test(vec2, conf.level = 0.95)$conf.int
ci2
## [1] 5.633699 5.737926
## attr(,"conf.level")
## [1] 0.95
ci2[1]
## [1] 5.633699

Q3

vec3 <- sample(vec1, 50, replace = FALSE)
avg3 <- mean(vec3)
avg3
## [1] 5.727459
ci3 <- t.test(vec3, conf.level = 0.95)$conf.int
ci3
## [1] 5.680356 5.774562
## attr(,"conf.level")
## [1] 0.95
ci3[1]
## [1] 5.680356

Q4

vec4 <- rnorm(1000, 5.7, 2)
avg4 <- mean(vec4)
plot(density(vec4))
abline(v = avg4, col = "red")

Q5

vec5 <- sample(vec4, 50, replace = FALSE)
avg5 <- mean(vec5)
avg5
## [1] 5.622747
ci5 <- t.test(vec5, conf.level = 0.95)$conf.int
ci5
## [1] 5.073613 6.171880
## attr(,"conf.level")
## [1] 0.95
ci5[2]
## [1] 6.17188

Q6

vec6 <- sample(vec4, 50, replace = FALSE)
avg6 <- mean(vec6)
avg6
## [1] 5.792576
ci6 <- t.test(vec6, conf.level = 0.95)$conf.int
ci6
## [1] 5.193495 6.391657
## attr(,"conf.level")
## [1] 0.95
ci6[2]
## [1] 6.391657

Code from Class

This demonstrates the meaning of confidence intervals

# Code for what we did Thursday: 
#simulate pop 
#rexp(number, rate for the exponentiale^x)
pop<-rexp(n=5000, .1)
hist(pop)
mean(pop)
plot(density(pop))
# visualize mean 
abline(v=mean(pop), col="red")
### sample from population how often get real mean
# writing a loop how many times do
## expiramenr 
reps<-3000
ssize<-1000
includes.true.value<-rep(F,reps)
means<-c()
for(i in 1:reps) {
  samp<- sample(pop, size = ssize)
  means[i] <- mean(samp)
  rang<- t.test(samp, conf.level=.95)$conf.int
  if(rang[1]< mean(pop) &
     mean(pop)< rang[2]) {
    includes.true.value[i]<-T
  }
  
}
sum(includes.true.value)/length(includes.true.value)
plot(density(means), xlim=c(0,max(pop)), col= "red")
lines(density(pop))
abline(v=mean(pop), col="blue")