install.packages("rmarkdown")

## Error in contrib.url(repos, "source"): trying to use CRAN without setting a mirror

install.packages("knitr")

## Error in contrib.url(repos, "source"): trying to use CRAN without setting a mirror

library(rmarkdown)
library(knitr)
Task 1 
#1. Calculate the mean and variance of the population.
population = c(0, 1, 4, 5)
mean_pop = mean(population)
var_pop = var(population)
mean_pop

## [1] 2.5

var_pop

## [1] 5.666667

#2. Sampling without replacement:
samples_wo = combn(population, 2)
sample_means_wo = colMeans(samples_wo)
expected_value_wo = mean(sample_means_wo)
variance_wo = var(sample_means_wo)
expected_value_wo

## [1] 2.5

variance_wo

## [1] 1.7

#3. Sampling with replacement:
samples_w = expand.grid(population, population)
sample_means_w = rowMeans(samples_w)
expected_value_w = mean(sample_means_w)
variance_w = var(sample_means_w)
expected_value_w

## [1] 2.5

variance_w

## [1] 2.266667
Task 2 
### Simulate 100 Observations:
set.seed(123)
n = 100
min_val = 0
max_val = 75
sample = runif(n, min_val, max_val)

# Estimators
T1 = 2 * mean(sample)
T2 = max(sample)

# Sequence of estimators
T1_seq = 2 * cumsum(sample) / (1:n)
T2_seq = cummax(sample)

# Plot
plot(1:n, T1_seq, type = 'l', col = 'blue', ylim = range(c(T1_seq, T2_seq, max_val)), ylab = "Estimator Values", xlab = "Sample Size")
lines(1:n, T2_seq, col = 'red')
abline(h = max_val, col = 'green', lty = 2)
legend("bottomright", legend = c("T1", "T2", "True Value"), col = c("blue", "red", "green"), lty = 1)
plot of chunk unnamed-chunk-3
### Simulation of 10,000 Samples:

simulations = 10000
T1_vals = numeric(simulations)
T2_vals = numeric(simulations)

for (i in 1:simulations) {
  sample = runif(n, min_val, max_val)
  T1_vals[i] = 2 * mean(sample)
  T2_vals[i] = max(sample)
}

# Boxplot
boxplot(T1_vals, T2_vals, names = c("T1", "T2"), main = "Boxplot of Estimators", col = c("blue", "red"))
plot of chunk unnamed-chunk-3
Task 3 
### Simulate 15 Observations from a Standard Normal Distribution:
set.seed(123)
n = 15
sample = rnorm(n)
alpha = 0.05

# Testing H0: mu = 0
t_test_mu_0 = t.test(sample, mu = 0)
t_test_mu_0

## 
##  One Sample t-test
## 
## data:  sample
## t = 0.69816, df = 14, p-value = 0.4965
## alternative hypothesis: true mean is not equal to 0
## 95 percent confidence interval:
##  -0.3157490  0.6205177
## sample estimates:
## mean of x 
## 0.1523843

# p-value
p_value_mu_0 = t_test_mu_0$p.value
p_value_mu_0

## [1] 0.4965107

# Repeat 10,000 times for H0: mu = 0
simulations = 10000
p_values_mu_0 = numeric(simulations)
for (i in 1:simulations) {
  sample = rnorm(n)
  p_values_mu_0[i] = t.test(sample, mu = 0)$p.value
}

# Simulated alpha level
alpha_simulated = mean(p_values_mu_0 < alpha)

# Histogram
hist(p_values_mu_0, main = "Histogram of p-values (H0: mu = 0)", col = "blue")
plot of chunk unnamed-chunk-4
alpha_simulated

## [1] 0.0473

### Testing H0: mu = 1
t_test_mu_1 = t.test(sample, mu = 1)
t_test_mu_1

## 
##  One Sample t-test
## 
## data:  sample
## t = -2.3729, df = 14, p-value = 0.03251
## alternative hypothesis: true mean is not equal to 1
## 95 percent confidence interval:
##  -0.3973286  0.9294368
## sample estimates:
## mean of x 
## 0.2660541

# p-value
p_value_mu_1 = t_test_mu_1$p.value
p_value_mu_1

## [1] 0.03250973

# Repeat 10,000 times for H0: mu = 1
p_values_mu_1 = numeric(simulations)
for (i in 1:simulations) {
  sample = rnorm(n)
  p_values_mu_1[i] = t.test(sample, mu = 1)$p.value
}

# Simulated power of the test
power_simulated = mean(p_values_mu_1 < alpha)

# Histogram
hist(p_values_mu_1, main = "Histogram of p-values (H0: mu = 1)", col = "red")
plot of chunk unnamed-chunk-4
power_simulated

## [1] 0.9493
Task 4 
### Simulate 15 Observations from a Log-Normal Distribution:
set.seed(123)
n = 15
mu = 1
sigma = 1.5
sample = rlnorm(n, meanlog = mu, sdlog = sigma)

# 95% Confidence Interval
alpha = 0.05
sample_mean = mean(sample)
sample_sd = sd(sample)
se = sample_sd / sqrt(n)
ci = sample_mean + c(-1, 1) * qt(1 - alpha/2, df = n - 1) * se

# Display results
sample_mean

## [1] 7.49678

ci

## [1]  1.531404 13.462157

### Repeat 10,000 Times:
simulations = 10000
contains_true_value = numeric(simulations)
below_true_value = numeric(simulations)
above_true_value = numeric(simulations)

for (i in 1:simulations) {
  sample = rlnorm(n, meanlog = mu, sdlog = sigma)
  sample_mean = mean(sample)
  sample_sd = sd(sample)
  se = sample_sd / sqrt(n)
  ci = sample_mean + c(-1, 1) * qt(1 - alpha/2, df = n - 1) * se
  true_value = exp(mu + (sigma^2) / 2)
  contains_true_value[i] = true_value >= ci[1] && true_value <= ci[2]
  below_true_value[i] = true_value < ci[1]
  above_true_value[i] = true_value > ci[2]
}

# Results
mean(contains_true_value)

## [1] 0.7471

mean(below_true_value)

## [1] 2e-04

mean(above_true_value)

## [1] 0.2527