LIS 4273 Module #6 Assignment

 #1 Ice Cream Purchases Population

#A Define the population

population <- c(8, 14, 16, 10, 11)

#B Calculate the mean

population_mean <- mean(population)

population_mean

[1] 11.8

#C Select a random sample of size 2

sample_size_2 <- sample(population, 2)

sample_size_2

[1] 14 10

# Calculate the sample mean

sample_mean <- mean(sample_size_2)

sample_mean

[1] 12

# Calculate the sample standard deviation

sample_sd <- sd(sample_size_2)

sample_sd

[1] 2.828427

#D Calculate the population standard deviation

population_sd <- sd(population)

population_sd

[1] 3.193744

#2 Sampling Distribution of Proportion

Does the sample proportion p have approximately a normal distribution? Explain.

Since both $np$ and $nq$ is greater than 5, the sampling distribution of $p$ will be approximately normal.

What is the smallest value of n for which the sampling distribution of p is approximately normal? 100

# Given values

n <- 100

p <- 0.95

q <- 1 - p

# Check np and nq

np <- n * p

nq <- n * q

np

[1] 95

nq

[1] 5

# Solve for minimum n where np and nq are >= 5

n_min <- 5 / min(p, q)

n_min

[1] 100

#3 Simulated Coin Tossing

Simulated coin tossing: is probability better done using function called rbinom than using function called sample?

In R, the rbinom() function simulates binomial distributions, which is better suited for probability experiments than sample() because rbinom() accounts for the number of trials and the probability of success directly.

# Simulate 100 coin tosses with a fair coin (p = 0.5)

coin_tosses_rbinom <- rbinom(100, size = 1, prob = 0.5)

table(coin_tosses_rbinom)

0 1

49 51

# Simulate 100 coin tosses using sample function

coin_tosses_sample <- sample(c(0, 1), 100, replace = TRUE, prob = c(0.5, 0.5))

table(coin_tosses_sample)

0 1

55 45