📘 How to Use the Population vs Sample Regression App

🎯 Objective

This app helps you visually understand the difference between three concepts:

• Data generating process (DGP)
• Population regression line (PRL)
• Sample regression line (SRL)

📝 What is shown?

The left-hand side graph shows a cloud of points generated by the next DGP:

\[ y_i = \beta_0 + \beta_1 x_i + \varepsilon_i, \quad \varepsilon_i \sim \mathcal{N}(0, \sigma^2) \]

Where: $$\beta_0 = 3,\; \beta_1 = 2,\; and \; \sigma^2=4$$ Together with its PRL in white:

\[ \mathbb{E}\left[y_i|x_i\right] = 3 + 2x_i \]

Sampled points are colored in red, with its corresponding SPL:

\[ \widehat{y_i} = \widehat{\beta_0} + \widehat{\beta_1}x_i \]

The right-hand side graph shows the beta estimators from the sample of points in the left-hand side as a function of the sample size.

🧪 Step-by-Step

Try the following steps:

1. Click the ▶ Animate button :
   • Left: watch how is filled with red dots while the red line gest closer to the white line.
   • Right: similarly, watch how the estimated beta in the sample gets closer to the true beta.
2. Slider: Adjust the sample size to update both plots.

💡 Learning Goals

• Differentiate between sample-based inference and population parameters.
• Get a sense on how the law of large numbers work.

🏁 Chalenge

Why is the estimated beta not equal to the true beta when we sample the whole population?

📖 Show Solution