Flipped_Assignment_13

Question 1

A sample of size n is to be drawn from a population with a known mean of 25.4 and a standard deviation of 2.8. Using the Central Limit Theorem.

1. What is the probability that the average of the n observations is less than 25.3 when: n=10, n=25, n=50?

# set parameters
mu <- 25.4
sigma <- 2.8
n1 <- 10
n2 <- 25
n3 <- 50
# get probability numbers for CLT
mu_xbar <- mu
sigma1 <- sigma /sqrt(n1)
sigma2 <- sigma /sqrt(n2)
sigma3 <- sigma /sqrt(n3)
# probabilities
pnorm(25.3, mean = mu_xbar, sd = sigma1)

## [1] 0.4550397

pnorm(25.3, mean = mu_xbar, sd = sigma2)

## [1] 0.4291371

pnorm(25.3, mean = mu_xbar, sd = sigma3)

## [1] 0.4003126

2. Comment on anything you notice about Pr(X < 25.3) as n is increased.

As n increased the probability that X < 25.3 decreased over that same span.

3. Plot on the same graph the pdf of X over the range X = (24,27) for the different sample sizes n.

curve(dnorm(x, mean = mu_xbar, sd = sigma1), from=24, to=27, col="blue", main="Pr(X<25.3) for n=10,25,50", xlab="X", ylab="Probability")
curve(dnorm(x, mean = mu_xbar, sd = sigma2), from=24, to=27, col="red", add=TRUE)
curve(dnorm(x, mean = mu_xbar, sd = sigma3), from=24, to=27, col="green", add=TRUE)

Question 2

A sample of n = 30 observations is to be drawn from a Binonmial(100, .05) distribution, each Xi ~ Bin(100, .05).

nbin <- 100
p <- 0.05
n <- 30
# a) mean of Xi
mu_xi <- nbin * p
print(mu_xi)

## [1] 5

# b) standard deviation of Xi
sd_xi <- sqrt(nbin * p * (1-p))
print(sd_xi)

## [1] 2.179449

# c) mean of X
mu_X <- mu_xi
print(mu_X)

## [1] 5

# d) standard deviation of X
sd_X <- sd_xi/sqrt(30)
print(sd_X)

## [1] 0.3979112

# e) Pr(Xi <= 4)
pbinom(4,n,p)

## [1] 0.9843645

# f) Pr(X <= 4)
pnorm(4,mu_X,sd_X)

## [1] 0.005983373

Complete Code

# set parameters
mu <- 25.4
sigma <- 2.8
n1 <- 10
n2 <- 25
n3 <- 50
# get probability numbers for CLT
mu_xbar <- mu
sigma1 <- sigma /sqrt(n1)
sigma2 <- sigma /sqrt(n2)
sigma3 <- sigma /sqrt(n3)
# probabilities
pnorm(25.3, mean = mu_xbar, sd = sigma1)

## [1] 0.4550397

pnorm(25.3, mean = mu_xbar, sd = sigma2)

## [1] 0.4291371

pnorm(25.3, mean = mu_xbar, sd = sigma3)

## [1] 0.4003126

curve(dnorm(x, mean = mu_xbar, sd = sigma1), from=24, to=27, col="blue", main="Pr(X<25.3) for n=10,25,50", xlab="X", ylab="Probability")
curve(dnorm(x, mean = mu_xbar, sd = sigma2), from=24, to=27, col="red", add=TRUE)
curve(dnorm(x, mean = mu_xbar, sd = sigma3), from=24, to=27, col="green", add=TRUE)

nbin <- 100
p <- 0.05
n <- 30
# a) mean of Xi
mu_xi <- nbin * p
print(mu_xi)

## [1] 5

# b) standard deviation of Xi
sd_xi <- sqrt(nbin * p * (1-p))
print(sd_xi)

## [1] 2.179449

# c) mean of X
mu_X <- mu_xi
print(mu_X)

## [1] 5

# d) standard deviation of X
sd_X <- sd_xi/sqrt(30)
print(sd_X)

## [1] 0.3979112

# e) Pr(Xi <= 4)
pbinom(4,n,p)

## [1] 0.9843645

# f) Pr(X <= 4)
pnorm(4,mu_X,sd_X)

## [1] 0.005983373