Muddy Points

Lesson 14: Hypothesis testing part 03

Modified

November 12, 2025

Fall 2025

1. How to rename datasets in R (is it just name2 = name1 or do you need to use a specific function/package?)

Yes, you can simply rename a dataset in R by assigning it to a new variable name using the assignment operator <- or =.

For example:

new_dataset <- old_dataset

or

new_dataset = old_dataset

This creates a new variable new_dataset that contains the same data as old_dataset.

2. Understanding how the means of the two samples fall on one distributive curve (still a bit confused)

The confusion about how two sample means fall on one curve is resolved by understanding that the “one curve” isn’t about the individual sample means (\(\bar{X}_1\) and \(\bar{X}_2\)), but about the difference between them (\(\bar{X}_1 - \bar{X}_2\)).

The single distribution you are looking for is the Sampling Distribution of the Difference in Sample Means.

The distribution of \(\bar{X}_1 - \bar{X}_2\) is a Normal distribution with:

  • Mean: \(E[\overline{X}_1 - \overline{X}_2] = \mu_1 - \mu_2\)
  • Standard Deviation (Standard Error): \(SD(\overline{X}_1 - \overline{X}_2) = \sqrt{\frac{\sigma_1^2}{n_1}+\frac{\sigma_2^2}{n_2}}\)

\[\overline{X}_1 - \overline{X}_2 \sim \text{Normal} \left( \mu_1 - \mu_2, \sqrt{\frac{\sigma_1^2}{n_1} + \frac{\sigma_2^2}{n_2}} \right)\]

Let’s look at the distributions

We’ll use R and ggplot2 to simulate two hypothetical populations and show the three key distributions.

1. R Setup and Parameters

We define the true population means (\(\mu\)) and standard deviations (\(\sigma\)) and a sample size (\(n\)).

2. Visualize the Individual Sample Mean Distributions.

This plot shows the two separate sampling distributions for \(\overline{X}_A\) and \(\overline{X}_B\).

Code 
# Data for the individual sampling distributions (for plot limits)
data_sd_A <- data.frame(mean_A = c(mu_A - 3*sigma_A/sqrt(n_A), mu_A + 3*sigma_A/sqrt(n_A)))
data_sd_B <- data.frame(mean_B = c(mu_B - 3*sigma_B/sqrt(n_B), mu_B + 3*sigma_B/sqrt(n_B)))

# --- Individual Sample Mean Distributions Plot ---
ggplot(data.frame(x = c(95, 110)), aes(x)) +
  # Distribution A (Centered at mu_A = 100)
  stat_function(fun = dnorm, args = list(mean = mu_A, sd = sigma_A / sqrt(n_A)), 
                geom = "area", fill = "blue", alpha = 0.4) +
  # Distribution B (Centered at mu_B = 105)
  stat_function(fun = dnorm, args = list(mean = mu_B, sd = sigma_B / sqrt(n_B)), 
                geom = "area", fill = "red", alpha = 0.4) +
  labs(title = "Distributions of Individual Sample Means",
       subtitle = expression(paste(bar(X)[A], " (Blue) and ", bar(X)[B], " (Red)")),
       x = "Sample Mean Value", y = "Density") +
  theme_minimal()

3. Take One Pair of Random Samples

We simulate the result of one experiment by taking one pair of samples.

# Take one random sample from each hypothetical population
sample_A <- rnorm(n = n_A, mean = mu_A, sd = sigma_A)
sample_B <- rnorm(n = n_B, mean = mu_B, sd = sigma_B)

# Calculate the single observed sample means and their difference
x_bar_A <- mean(sample_A)
x_bar_B <- mean(sample_B)
observed_diff <- x_bar_A - x_bar_B
  • One Observed Sample Mean A (x_bar_A): 100.69
  • One Observed Sample Mean B (x_bar_B): 104.02
  • Observed Difference (x_bar_A - x_bar_B): -3.34

4. Visualize the Sampling Distribution of the Difference

This is the single curve where the observed difference from our samples falls. It’s centered at the true difference in population means (\(\mu_A - \mu_B = -5\))

Code 
# --- Sampling Distribution of the DIFFERENCE Plot ---
max_density <- dnorm(true_diff_mean, true_diff_mean, sd_diff_mean)

ggplot(data.frame(x = c(true_diff_mean - 4*sd_diff_mean, true_diff_mean + 4*sd_diff_mean)), aes(x)) +
  # The Distribution of the DIFFERENCE (The "One Curve")
  stat_function(fun = dnorm, args = list(mean = true_diff_mean, sd = sd_diff_mean), 
                geom = "area", fill = "purple", alpha = 0.6) +
  
  # Line for the True Difference (Center of the distribution)
  geom_vline(xintercept = true_diff_mean, color = "black", linetype = "dashed", size = 1) +
  
  # Line for the ONE Observed Difference from our samples
  geom_vline(xintercept = observed_diff, color = "darkgreen", size = 1.2) +
  
  labs(title = "The Sampling Distribution of the DIFFERENCE in Means",
       subtitle = "The 'One Curve' for the Difference in Sample Means",
       # SIMPLIFIED X-AXIS LABEL (This is the stable fix)
       x = "Difference in Sample Means (Xbar_A - Xbar_B)", 
       y = "Density") +
  
  # Text for True Difference 
  geom_text(aes(x = true_diff_mean, y = max_density * 0.95), 
            label = "True Difference: -5", 
            hjust = 1.1, size = 4) +
  
  # Text for Observed Difference 
  geom_text(aes(x = observed_diff, y = max_density * 0.85), 
            label = paste("Observed Difference:", round(observed_diff, 2)), 
            hjust = -0.1, size = 4, color = "darkgreen") +
  
  theme_minimal()

The Takeaway

The final graph demonstrates the core concept:

  1. We collect two samples and calculate their means, \(\bar{X}_A\) and \(\bar{X}_B\).
  2. We compute the single value of the observed difference: \(\bar{X}_A - \bar{X}_B\) (the dark green line).
  3. This single value is positioned on the single distribution (the purple curve), which represents the likelihood of obtaining any possible difference, assuming the true difference is \(\mu_A - \mu_B\) (the dashed black line).