Chapter 9 Page 354 Exercise 1.

  1. A die is rolled 24 times. Use the Central Limit Theorem to estimate the probability that
  1. the sum is greater than 84.
  2. the sum is equal to 84.

Ans. (a) the sum is greater than 84.

## EV = Expected value
## VX = Variance
## SD = Standard Deviation

EVX <- (1 * (1/6) + 2 * (1/6) + 3 * (1/6) + 4 * (1/6) + 5 * (1/6) + 6 * (1/6))
EVX
## [1] 3.5
VX <- 1/6 * ((1 - EVX)^2 + (2 - EVX)^2 + (3 - EVX)^2 + (4 - EVX)^2 + (5 - EVX)^2 + (6 - EVX)^2)
VX
## [1] 2.916667
EV24 = 24 * EVX
EV24
## [1] 84
V24 = 24 * VX
V24
## [1] 70
SD24 <- sqrt(V24)
SD24
## [1] 8.3666
P <- (84.5 - EV24)/SD24
P
## [1] 0.05976143
PS24 <- round(1 - pnorm(P),4)
PS24
## [1] 0.4762

The probability that the sum is greater than 84 is 0.4762

  1. the sum is equal to 84.
## Sum not equal to 84 is equal to higher than 84 or lower than 84
Phigher <- (84.5)
Plower <- (83.5)

  
PEQ_84 <- round(pnorm(84.5, mean = 84, SD24)- pnorm(83.5, mean = 84, SD24),4) 
PEQ_84
## [1] 0.0477

The probability that the sum is equal to 84 is 0.0477