sol2

x <- 0:6
freq <- c(0,4,19,15,17,7,2)
barplot(freq)

x.mean <- sum(x*freq)/sum(freq)
x.mean

## [1] 3.15625

x.var <- sum((x-x.mean)^2*freq)/sum(freq-1)
x.var

## [1] 1.691886

z.test <- (x.mean-2)/sqrt(x.var)

p.value <- 1-pnorm(0.94)

p.value

## [1] 0.1736088

varSimu<-{}
N=1000
for (i in 1:N){
  sample = rbinom(64,6,1/3)
  #rbinom(n, size, prob)
  varSimu <- c(varSimu, var(sample))
}

p.value.Simu <- sum(varSimu>x.var)/N

hist(varSimu, main= paste("Based on resampling, pvalue =",round( p.value.Simu,3)))
abline( v=x.var, col ="red")

probs <- c( dbinom(0,6,1/3),
dbinom(1,6,1/3),
dbinom(2,6,1/3),
dbinom(3,6,1/3),
dbinom(4,6,1/3),
dbinom(5,6,1/3),
dbinom(6,6,1/3))
64*probs

## [1]  5.6186557 16.8559671 21.0699588 14.0466392  5.2674897  1.0534979
## [7]  0.0877915