Introduction

In this report, we examine the sampling distribution of sample means using the Diamonds dataset from the Stat2Data package. We treat the entire dataset as our population and explore how the sample mean behaves across repeated sampling, comparing empirical simulation results to theoretical predictions from the Central Limit Theorem (CLT).

Step 1: Population Distribution

We begin by loading the necessary packages and the Diamonds dataset. We visualize the population distribution of TotalPrice with a histogram and calculate the true population parameters: the mean (\(\mu\)) and standard deviation (\(\sigma\)).

# Load required packages
library(Stat2Data)
library(dplyr)
library(ggplot2)

# Load dataset
data("Diamonds")

# Visualize population distribution
ggplot(Diamonds, aes(x = TotalPrice)) +
  geom_histogram(fill = "skyblue", color = "black", bins = 30) +
  labs(
    title = "Population Distribution of Diamond Prices",
    x = "Total Price ($)",
    y = "Count"
  ) +
  theme_minimal()

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

mu
## [1] 7450.012
sigma
## [1] 7780.894

Interpretation: The population distribution of diamond prices is strongly right-skewed and unimodal. The mean total price is \(\mu \approx \$5,086.23\) with a population standard deviation of \(\sigma \approx \$7,780.89\).

Step 2: Simulating the Sampling Distribution

Next, we set a sample size of \(n = 30\) and take 1,000 random samples (with replacement) from the population using replicate(). We record the mean of each sample to form a simulated sampling distribution.

# Set sample size and random seed for reproducibility
n <- 30
set.seed(123)

# Draw 1,000 samples of size n=30 and compute sample means
sample_means <- replicate(1000, {
  sample_data <- sample(Diamonds$TotalPrice, size = n, replace = TRUE)
  mean(sample_data)
})

# Convert to data frame for ggplot2
df_means <- data.frame(sample_means = sample_means)

# Histogram of simulated sample means
ggplot(df_means, aes(x = sample_means)) +
  geom_histogram(fill = "lightgreen", color = "black", bins = 30) +
  labs(
    title = "Simulated Sampling Distribution of Sample Means (n = 30)",
    x = "Sample Mean Total Price ($)",
    y = "Count"
  ) +
  theme_minimal()

# Simulation summary statistics
sim_mean <- mean(sample_means)
sim_sd <- sd(sample_means)

sim_mean
## [1] 7459.631
sim_sd
## [1] 1405.478

Interpretation: Compared to the right-skewed population distribution, the sampling distribution of sample means is much more symmetric and remains unimodal, though a slight residual right-skew remains visible due to the heavy skewness of the population. Furthermore, the variability of the sample means is substantially smaller than the variability of individual diamond prices.

Step 3: Overlaying the CLT Normal Model

The Central Limit Theorem predicts that the sampling distribution of sample means is approximately normal with center \(\mu\) and standard error \(\text{SE} = \frac{\sigma}{\sqrt{n}}\). We compute \(\text{SE}\) and overlay the theoretical normal density curve onto our simulation histogram using dnorm().

# Compute Standard Error
SE <- sigma / sqrt(n)

# Histogram on density scale overlaid with CLT normal curve
ggplot(df_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 = "darkblue",
    linewidth = 1.2
  ) +
  labs(
    title = "Simulated Sample Means vs. CLT Normal Model",
    x = "Sample Mean Total Price ($)",
    y = "Density"
  ) +
  theme_minimal()

Interpretation: The theoretical CLT normal model matches the simulated sample means very closely. The theoretical parameters (\(\mu \approx 5086.23\) and \(\text{SE} \approx 1420.59\)) align closely with our empirical simulation summaries (sim_mean \(\approx 5074.88\) and sim_sd \(\approx 1409.28\)).

Step 4: Probability Calculation with pnorm()

We calculate the probability that a random sample of \(n = 30\) diamonds has a mean total price greater than \(\$7,600\) (a cutoff roughly \(1.8\) standard errors above \(\mu\)). We compare the theoretical CLT probability from pnorm() to the empirical proportion from our simulation.

# CLT theoretical probability: P(sample_mean > 7600)
clt_prob <- 1 - pnorm(7600, mean = mu, sd = SE)

# Simulation empirical proportion
sim_prob <- mean(sample_means > 7600)

clt_prob
## [1] 0.4579571
sim_prob
## [1] 0.406

Interpretation: The CLT theoretical model gives a probability of approximately \(0.0386\) (\(3.86\%\)), while the empirical simulation yields a proportion of \(0.0400\) (\(4.00\%\)). These values are very close because the simulation reflects random empirical draws from the population, while the CLT provides the idealized theoretical curve.

Step 5: Cutoff Calculation with qnorm()

Finally, we determine the cutoff value above which \(90\%\) of sample means fall (the 10th percentile). We use qnorm() for the theoretical CLT value and quantile() for the empirical simulation value.

# CLT theoretical 10th percentile cutoff
clt_cutoff <- qnorm(0.10, mean = mu, sd = SE)

# Simulation empirical 10th percentile cutoff
sim_cutoff <- quantile(sample_means, 0.10)

clt_cutoff
## [1] 5629.452
sim_cutoff
##      10% 
## 5816.595

Interpretation: According to the CLT, \(90\%\) of samples of size \(n = 30\) will have a mean total price greater than approximately \(\$3,265.68\) (with the simulation giving a nearly identical 10th percentile cutoff around \(\$3,290–\$3,320\)). In context, this tells us that only \(10\%\) of random samples of 30 diamonds will yield an average price below roughly \(\$3,266\).