Exhaustive Notes on Single Variable Inference, Randomization Tests, and Parameter Estimation

Course Mechanics and Exam Administration

• Exam Room Assignments and Rules:

  • Room assignments are strictly enforced based on class section numbers.

  • Sections 37 through 40 take their writing exam in the Engineering Building.

  • Section 46 and other sections have distinct assigned rooms.

  • Approximately half of the overall class is assigned to the Anthony building.

  • Proctors inside assigned exam rooms will refuse to attend to students who belong to a different section or room assignment to prevent unnecessary stress.

• Study Strategies and Grading Expectations:

  • Daily recommended study time is $30\,\text{minutes}$ to $1\,\text{hour}$ dedicated specifically to scoring and practicing exam concepts.

  • This unit's material represents the most accessible exam content to score high points on.

  • For students aiming for a $4.0$ GPA, scoring below $80\%$ on this exam threatens that GPA standing and risks dropping the overall grade toward a $3.5$

• Homework Integrity and Simulation Skills:

  • Active individual completion of daily homework assignments is required.

  • Exam questions directly reflect homework problems. Reliance on external help or automated tools without understanding underlying concepts will be obvious during individual exam performance.

  • Asking clarifying questions when receiving tutoring or peer guidance is critical to mastering the underlying foundational mechanics.

• Course Schedule and Software Assignments:

  • Top Hat questions will be posted late at night or early the following morning and will remain open until the following Wednesday.

  • Top Hat questions are specifically curated to prepare students for upcoming exam formats.

  • Randomization procedures constitute approximately $85\%$ of statistical course material moving forward until graduation.

Single Variable Analysis and Randomization Principles

• Transition in Statistical Focus:

  • Previous units evaluated the relationship between two variables.

  • Current focus shifts to analyzing one variable to measure whether a specific change (either an increase or a decrease) has occurred relative to a benchmark.

• Simulation and Randomization Methodology:

  • Single variable inference uses simulation-based randomization methods, specifically Bootstrap methods and sharp tests.

• Definitions of Categorical Outcomes:

  • A single categorical variable under study contains two mutually exclusive outcomes: Success and Failure.

  • Success and failure are generic statistical terms defined entirely by the experimenter or researcher.

  • Coin Toss Strategy Example: In a coin-tossing game between two players where Player A wins on Heads and Player B wins on Tails, landing on Heads represents a "success" for Player A and a "failure" for Player B.

Case Study 1: Facial Prototyping and Single Proportion Hypothesis Testing

• Facial Prototyping Concept:

  • Facial prototyping refers to the psychological phenomenon where individuals naturally associate specific facial expressions, visual structures, or characteristics with specific personal names.

• Classroom Data Collection:

  • Total sample size of participating students: $N = 272$

  • Experiment setup: Students chose between two facial images to identify which face belonged to the name "Tim".

  • Success count (students picking the left facial image as Tim): $k = 220$

  • Failure count (students picking the right facial image as Tim): $272 - 220 = 52$

• Hypotheses Formulation:

  • Research Question: Does facial appearance or social characteristic influence how people associate names with faces?

  • Null Hypothesis ($H_0$): Represents the statement of no change, no preference, or random guessing. The proportion of people picking the left face as Tim equals the proportion picking the right face as Tim.     $H_0: p = 0.50$

  • Alternative Hypothesis ($H_a$): Represents the statement that the preference or proportion differs from random chance.     $H_a: p \neq 0.50$

  • Directionality: Using the term "different" in the alternative hypothesis specifies a two-tailed test requiring evaluation in both tails of the distribution.

Physical Simulation Methodology Using Cards

• Card Deck Construction Rules:

  • To manually simulate a null distribution for a single proportion, always select an even number of physical playing cards representing the null ratio ($50\% / 50\%$).

  • Card selection uses two distinct colors (e.g., green and white).

  • Half of the cards are colored green to represent a "success" (e.g., selecting the face on the left as Tim).

  • Half of the cards are colored white (or red) to represent a "failure".

• Execution Step-by-Step Procedure:

  1. Take a physical deck containing an equal ratio of green and white cards (e.g., $100$ total cards or matching sample size $N = 272$).

  2. Thoroughly shuffle the deck to ensure complete randomization.

  3. Draw a sample equal to the study sample size ($N = 272$) with replacement.

  4. Record the simulated proportion of green cards ($\hat{p}_{sim}$).

  5. Repeat the shuffling, sampling, and proportion calculation process thousands of times (e.g., $5000$ iterations) to construct the empirical null distribution.

P-Value Estimation and Directionality Rules

• P-Value Mechanics and Tail Area Rules:

  • The total area under any probability distribution curve equals $100\%$ or $1.00$

  • Two-Tailed Test: Calculate the area to the right of the positive observed value PLUS the symmetrical area to the left of its negative counterpart (e.g., for an observed value at $0.60$, sum the area to the right of $0.60$ and the area to the left of $-0.60$).

  • Right-Tailed Test: Triggered by directional research keywords such as greater than, higher, superior, or bigger. Measure exclusively the tail area to the right of the observed value.

  • Left-Tailed Test: Triggered by directional research keywords such as lower, smaller, inferior, or less than. Measure exclusively the tail area to the left of the observed value.

• Visual P-Value Estimation Technique:

  • If an observed sample value isolates approximately $20\%$ of the distribution curve's total area in the tail, the estimated p-value is approximately $0.20$

  • Compare visual estimates directly against standard significance thresholds (e.g., $\alpha = 0.05$, $\alpha = 0.01$, $\alpha = 0.10$).

• Data Analysis and P-Value Result for Case Study 1:

  • Sample Size ($N$): $272$

  • Observed Successes ($k$): $220$

  • Observed Sample Proportion ($\hat{p}$):     $\hat{p} = \frac{220}{272} \approx 0.8088$

  • Claimed Null Parameter ($p_0$): $0.50$

  • Number of Computer Simulations: $5000$

  • Calculated P-Value: $0.0000$

  • Statistical Conclusion: A p-value of $0.0000$ provides extremely strong evidence against the null hypothesis ($H_0$). The null parameter of $50\%$ is entirely incompatible with the observed class data. It is conclusively demonstrated that people systematically associate specific names with specific facial characteristics.

Case Study 2: Vaping Usage and Claimed Parameters

• Distinguishing Parameters vs. Statistics vs. Claimed Parameters:

  • Parameter: A numerical summary or characteristic that describes an entire population.

  • Statistic: A numerical summary or characteristic calculated directly from a sample.

  • Claimed Parameter: A theoretical or historically accepted population parameter value assumed to be true prior to empirical investigation (e.g., census benchmark estimates).

• Study Context and Data:

  • Historical US Census Bureau benchmark data estimates that $22\%$ of all US adults use e-cigarette/vaping products regularly.

  • Population of Interest: All US adults aged 18 through 25.

  • Claimed Parameter ($p_0$): $p_0 = 0.22$ ($22\%$).

  • Research Question: Is the proportion of young adults aged 18 through 25 who use these products different from the general US adult population benchmark ($22\%$)?

  • Sample Data Collected:

  • Sample size ($n$): $500$ US adults aged 18 through 25.

  • Number of affirmative users ($k$): $244$

  • Observed Sample Statistic ($\hat{p}$):       $\hat{p} = \frac{244}{500} = 0.488$

• Card Simulation Setup for Claimed Parameter $p_0 = 0.22$:

  • Total physical deck size: $100$ cards.

  • Green cards (representing product users): $22$ cards ($22\%$).

  • White cards (representing non-users): $78$ cards ($78\%$).

  • Draw samples of $n = 500$ with replacement, calculating simulated sample proportions over repeated iterations to build the null distribution.

Centering Rules for Null Distributions

• Comparison of One-Variable vs. Two-Variable Centering:

  • Two-Variable Null Distribution: When analyzing the relationship or difference between two groups (e.g., $\hat{p}_1 - \hat{p}_2$ or $\bar{x}_1 - \bar{x}_2$), the null distribution is always centered at $0$.

  • Single-Variable Null Distribution: When analyzing a single categorical variable proportion, the null distribution is always centered at the claimed parameter ($p_0$).

  • In the vaping study where the claimed parameter is $22\%$ ($p_0 = 0.22$), the symmetric bell-shaped null distribution has its peak/center located precisely at $0.22$, not at $0$

Student Dialogue and Peer Discussions

• Discussion on Assignments and Homework Schedule:

  • Homework 4 was previously due; subsequent assignment deadlines were adjusted and moved to accommodate upcoming exam schedules.

  • Students noted that working through practical homework problems provided greater learning utility than listening to passive lecture theory alone.

• Student Commentary on Campus Clubs and Social Events:

  • Students discussed campus involvement, career fairs, and local club activities.

  • Specific peer exchanges involved student involvement in Supply Chain Management clubs, wakeboarding, and skiing activities.

  • Lighthearted conversations touched upon local campus hazing rumors, including anecdotes regarding students locked in classrooms near cornfields.