Lecture 15 – Models and Viewpoints

DSC 10, Fall 2022

Announcements

Agenda

Statistical models

Models

Example

Galileo's Leaning Tower of Pisa Experiment

Example: Jury selection

Swain vs. Alabama, 1965

$\substack{\text{eligible} \\ \text{population}} \xrightarrow{\substack{\text{representative} \\ \text{sample}}} \substack{\text{jury} \\ \text{panel}} \xrightarrow{\substack{\text{selection by} \\ \text{judge/attorneys}}} \substack{\text{actual} \\ \text{jury}}$

Supreme Court ruling

"... the overall percentage disparity has been small...”

Our model for simulating Swain's jury panel

Our approach: simulation

Simulating statistics

Recall, a statistic is a number calculated from a sample.

  1. Run an experiment once to generate one value of a statistic.
    • In this case, sample 100 people randomly from a population that is 26% Black, and count the number of Black men (statistic).
  1. Run the experiment many times, generating many values of the statistic, and store these statistics in an array.
  1. Visualize the resulting empirical distribution of the statistic.

Step 1 – Running the experiment once

np.random.multinomial(sample_size, pop_distribution)

Aside: Example usage of np.random.multinomial

Halloween is on Monday, and you're getting ready to go trick-or-treating 👻. Suppose you'll visit 35 houses, and that each of the 35 houses you'll visit has the same candy box, containing:

At each house, you'll select one candy blindly from the candy box.

To simulate the act of going to 35 houses, we can use np.random.multinomial:

Step 1 – Running the experiment once

In our case, a randomly selected member of our population is Black with probability 0.26 and not Black with probability 1 - 0.26 = 0.74.

Each time we run the following cell, we'll get a new random sample of 100 people from this population.

Step 1 – Running the experiment once

We also need to calculate the statistic, which in this case is the number of Black men in the random sample of 100.

Step 2 – Repeat the experiment many times

Step 3 – Visualize the resulting distribution

Was a jury panel with 8 Black men suspiciously unusual?

Conclusion

Example: Genetics of peas 🟢

Gregor Mendel, 1822-1884