# Simulation in R for Sampling (2024)

The post is about simulation for sampling in R Programming Language. It contains some useful basic examples for generating samples and then computing some basic calculations in generated data.

Question 1: Simulate a coin toss 20 times.

sample(c("H", "T"), 20, replace=T)

Question 2: Write R commands to find out the 95% confidence interval for the mean (unknown variance) from the following population

yp <- c(111, 150, 121, 198, 112, 136, 114, 129, 117, 115, 186, 110, 121, 115, 114)
N  <- length(yp)
ys <- sample(yp, 5)
n  <- length(ys)
mys <- mean(ys)
vys <- vary(ys)
vybar <- var(yp)/n
sdr <- sqrt(vybar)
error <- qnorm(0.975)*sdr
ll <- mys - error
ul <- mys + error

### Sampling without Replacement and Histogram

Question 3: If we have a population ِye <- c(112, 114, 119, 125, 158, 117, 135, 141, 185, 128) then simulate this population with $k=100$ and $n=3$ for Simple Random Sampling without Replacement (SRSWOR). Also, find out the sample mean. Draw the histogram of the sample means generated.

k = 100; n = 3
m1 <- c()
ye <- c(112, 114, 119, 125, 158, 117, 135, 141, 185, 128)

for(i in 1:100){
s <- sample(ye, 3)
m1[i] <- mean(s)
}

m1
hist(m1)

Question 4: Perform a simulation in R by writing the R code considering generating a population of size 500 values from a normal distribution with a mean = 20 and a standard deviation = 30. Select 5000 samples, each of size 50 using the systematic sampling technique, and estimate the mean of each sample. Find the mean and variance of 5000 means.

N = 500; n = 50;
k = N/n; m = c();
pop <- rnorm (N, mean=20, sd=30)

for(i in 1:5000){
start <- sample(1: k, 1)
s <- seq(start, N, k)
sys.sample <- pop[s]
m[i] = mean(sys.sample)
}

mean(m); var(m)

Question 5: Why do we use simulation for sampling?
Answer: The simulation study is useful to evaluate a sampling strategy. We can generate the populations considering specific situations. Generating the population, the sample of size $n$ is obtained $k$ times. From each sample, the estimator is obtained. The variance of $k$ estimators is calculated for examining the efficiency.

### Coin Toss Experiment in R

Question 6: Write an R code to Simulate a coin-tossing experiment.

# Define the Number of tosses of a coin
n_tosses <- 100

# Simulate coin tosses (1 for heads, 0 for tails)
coin_tosses <- sample(c(0, 1), n_tosses, replace = TRUE)

# Calculate the proportion of heads

# Display results
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