Step 1: Population Parameters & Distribution

We examine the TotalPrice variable from the Diamonds dataset.

# Population parameters
mu <- mean(Diamonds$TotalPrice)
sigma <- sd(Diamonds$TotalPrice)

mu
## [1] 7450.012
sigma
## [1] 7780.894
# Population Histogram
ggplot(Diamonds, aes(x = TotalPrice)) +
  geom_histogram(fill = "skyblue", color = "black", bins = 30) +
  labs(title = "Population Distribution of Diamond Total Price",
       x = "Total Price ($)", y = "Count") +
  theme_minimal()

Interpretation: The population distribution of TotalPrice is strongly right-skewed with a mean of 7450.01 and standard deviation of 7780.89.

Step 2 & 3: Sampling Distribution Simulation & CLT Overlay

We simulate 1,000 random samples of size \(n = 30\) with replacement to construct the empirical sampling distribution of sample means, and overlay the theoretical CLT model \(N(\mu, SE)\).

set.seed(123)
n <- 30
num_sims <- 1000

# Simulate sampling distribution
sample_means <- replicate(num_sims, {
  sample_data <- sample(Diamonds$TotalPrice, size = n, replace = TRUE)
  mean(sample_data)
})

df_sample_means <- data.frame(sample_means)

# Theoretical SE
SE <- sigma / sqrt(n)

# Plot Sampling Distribution with CLT Normal Overlay
ggplot(df_sample_means, aes(x = sample_means)) +
  geom_histogram(aes(y = after_stat(density)), fill = "lightgreen", color = "black", bins = 30) +
  stat_function(fun = dnorm, args = list(mean = mu, sd = SE), color = "red", linewidth = 1) +
  labs(title = "Sampling Distribution vs. CLT Normal Model (n = 30)",
       x = "Sample Mean Total Price ($)", y = "Density") +
  theme_minimal()
## Warning: Ignoring unknown parameters: linewidth

# Summary statistics comparison
mean_sim <- mean(sample_means)
sd_sim <- sd(sample_means)

data.frame(
  Metric = c("Mean", "Standard Error / SD"),
  Theoretical_CLT = c(mu, SE),
  Simulated_Empirical = c(mean_sim, sd_sim)
)
##                Metric Theoretical_CLT Simulated_Empirical
## 1                Mean        7450.012            7459.631
## 2 Standard Error / SD        1420.590            1405.478

Interpretation: Despite the original population being right-skewed, the sampling distribution of sample means (\(n = 30\)) is nearly symmetric and bell-shaped, aligning closely with the CLT’s theoretical normal curve.

Step 4: Probability Analysis (\(P(\bar{X} > 10,000)\))

We calculate the probability that a sample mean exceeds \(10,000\).

cutoff <- 10000

# Theoretical CLT Probability
prob_clt <- 1 - pnorm(cutoff, mean = mu, sd = SE)

# Empirical Simulated Proportion
prop_sim <- mean(sample_means > cutoff)

c(Theoretical_CLT = prob_clt, Simulated_Empirical = prop_sim)
##     Theoretical_CLT Simulated_Empirical 
##          0.03632525          0.05000000

Interpretation: The theoretical probability of drawing a sample mean greater than \(10,000\) is approximately 0.0363. The simulated proportion (0.05) matches closely, with minor variation due to random sampling.

Step 5: Percentile Cutoff (10th Percentile)

We determine the price cutoff above which 90% of sample means lie (the 10th percentile).

# Theoretical 10th Percentile Cutoff
cutoff_clt <- qnorm(0.10, mean = mu, sd = SE)

# Empirical 10th Percentile Cutoff
cutoff_sim <- quantile(sample_means, 0.10)

c(Theoretical_CLT = cutoff_clt, Simulated_Empirical = cutoff_sim)
##         Theoretical_CLT Simulated_Empirical.10% 
##                5629.452                5816.595

Interpretation: According to the Central Limit Theorem, 90% of random samples of size \(n = 30\) will have an average total price of at least $5629.45.

Conclusion

This analysis demonstrates the Central Limit Theorem in action: 1. The sampling distribution of sample means approaches normality as sample size increases, even when drawn from a heavily skewed population. 2. The center of the sampling distribution approximates the population mean (\(\mu\)). 3. The variability of sample means is governed by the standard error (\(SE = \sigma / \sqrt{n}\)).