ERoman_Discussion9.rmd

Page 338 Exercise 1

Let S100 be the number of heads that turn up in 100 tosses of a fair coin. Use the Central Limit Theorem to estimate

1. P(S100 ≤ 45).

n<- 100
p<- 0.5
q<-1-p
mu <- n*p
sigma <- sqrt(n*p*q)
mu

## [1] 50

sigma 

## [1] 5

pu <-(45+1/2-50)/5
print(paste0("upper bound = ", pu))  

## [1] "upper bound = -0.9"

p <- pnorm(-0.9)
print(paste0("Estimate =: ", p))

## [1] "Estimate =: 0.18406012534676"

2. P(45 < S100 < 55).

#P(Sn > (45-1/2-50)/5)
pl <-(45-1/2-50)/5
print(paste0("lower bound = ", pl))

## [1] "lower bound = -1.1"

pu <-(55+1/2-50)/5
print(paste0("upper bound = ", pu))

## [1] "upper bound = 1.1"

p <- pnorm(1.1) - pnorm(-1.1)
print(paste0("Estimate =: ", p))

## [1] "Estimate =: 0.728667878107235"

3. P(S100 > 63).

pu <- (63+1/2-50)/5
print(paste0("upper bound = ", pu))

## [1] "upper bound = 2.7"

p <- 1- pnorm(2.7)
print(paste0("Estimate =: ", p))

## [1] "Estimate =: 0.00346697380304062"

4. P(S100 < 57).

pu <- (57+1/2-50)/5
print(paste0("upper bound = ", pu))

## [1] "upper bound = 1.5"

p <- pnorm(1.5)
print(paste0("Estimate =: ", p))

## [1] "Estimate =: 0.933192798731142"