Introduction to Statistical Reasoning Study Notes

Unit Overview and Administrative Information

  • Key Personnel:
    • Soo Jin: Chief Examiner, Lecturer, and Unit Coordinator.
    • Jesse: Unit Coordinator.
    • Contact: Questions, concerns, and personal issues should be directed via the provided email address. Non-personal questions should be posted to the Moodle forum so all students can benefit from the answers.
  • Unit Duration: The semester consists of 12 weeks of learning basic statistical methods and techniques.
  • General Definition of Statistics: Statistics is a science used to answer important questions, such as the average salary of a data analyst or whether drinking coffee improves memory.
  • Fields of Application: Business, finance, medical sciences, sports management, political sciences, social sciences, and technology (specifically involving high-dimensional and big data).

Learning Components and Structure

  • Statistical Branches:
    • Descriptive Statistics: Describing sample data numerically (mean, standard deviation) or graphically (histograms, box plots, scatter plots).
    • Inferential Statistics: Using sample information to make estimations or inferences about a general population.
  • Weekly Unit Schedule:
    • Part 1: Data Collection: Focuses on producing data through sampling designs and experimental designs.
    • Part 2: Descriptive Statistics: Covering numerical descriptions of single variables, relationships between two variables using scatter plots, and regression analysis.
    • Part 3: Basic Inference: Introduction to sampling distributions, the Central Limit Theorem, confidence intervals, and the logic behind hypothesis testing.
    • Part 4: Applied Inference: Applying inferential techniques to various scenarios, such as comparing two means or making inferences about proportions.
  • Learning Activities:
    • Moodle Own-time Videos: Explain key theories, terminology, and core concepts with minimal examples.
    • Seminars: Focus on practical examples, real-world applications, and walking through problems.
    • Applied Classes (Applied Workshop): Small group classes focusing on additional examples, step-by-step calculations, and hands-on practice using Excel with tutor assistance.
  • Attendance and Recordings:
    • Seminars and their recordings are available; recordings are typically posted 30 minutes after the session concludes.
    • Applied classes are in-person only and are not recorded.
    • There is a one-week delay in applied classes; for example, the concepts discussed in the Week 1 seminar are practiced in the Week 2 applied class.

Learning Resources and Support

  • Textbook: The Basic Practice of Statistics, 9th Edition. While the 9th edition is the most recent, older versions are acceptable for study purposes.
  • Math Learning Center (MLC): A drop-in help session service.
    • Hours: Monday to Friday, 11:00 a.m.11:00\,\text{a.m.} to 2:00 p.m.2:00\,\text{p.m.}.
    • Schedule: Opens in Week 2, including the mid-semester break, SWOTVAC, and the first two weeks of the exam period.
    • Location: First ground of the math building.
  • Peer Assisted Study Sessions (PASS): An academic mentoring program guided by students who previously completed the unit. Focuses on improving grades and sharing study tips. Sessions occur on Wednesdays and Fridays.
  • Disability Support Services: Provides support for students with disabilities, medical conditions, or mental health issues, including accommodations like extra exam time.

Assessment Summary

  • Applied Class Exercises:
    • 10 exercises in total; the best 8 results count toward the grade.
    • Completed on-campus during the specific applied class.
    • Access requires a passcode provided by the tutor at the end of the class.
  • Seminar Check-in Quizzes:
    • 11 quizzes total; the best 10 results count.
    • Opens Fridays at 5:30 p.m.5:30\,\text{p.m.} and closes Sunday at midnight.
    • Contains 5 simple questions (mix of multiple-choice and short-answer) based on seminar and pre-recorded material.
    • Only one attempt is allowed.
  • Review Quizzes:
    • Opens every Monday and closes Sunday at midnight.
    • Contains 10 multiple-choice questions based on the previous week's topic.
  • Statistical Analysis Project: A single, comprehensive project due on Friday of Week 10. Covers sampling, experimental design, graphical analysis, and inferential questions.
  • Mid-semester Test: Worth 20%20\% of the final grade. It is a closed-book test held in the Week 8 applied class (prior to mid-semester break).
  • Final Examination: Closed-book exam held during the exam period.
    • Equipment: Students must bring a scientific, non-programmable, and non-graphical calculator.
    • Approval: Calculators must have an official approval sticker.
  • Late Policy: A 5%5\% penalty applies for each calendar day late, including weekends.
  • Special Consideration: Applications must be submitted no later than midnight on the due date. Short extensions do not apply to applied class exercises; students must apply for special consideration to be exempt.

Sampling Methods

  • Simple Random Sample (SRS):
    • Assumes homogeneity in the population.
    • Excel Implementation:
      1. In the "Data" tab, select "Data Analysis" and "Random Number Generation."
      2. Set "Number of Variables" to 1 and "Number of Random Numbers" to the population size (e.g., 18).
      3. Set "Distribution" to "Uniform" between 0 and 1.
      4. Provide a "Random Seed" (any number, e.g., 20).
      5. Output the numbers into a column (e.g., Column D).
      6. Highlight the population list and the random numbers, then use the "Sort" function to sort by the random numbers (Smallest to Largest).
      7. Select the top nn units (e.g., the top 4) as the sample.
    • Drawback: It is theoretically possible to obtain an unrepresentative sample (e.g., all male or all female) by chance, which may bias results if that trait affects the response variable.
  • Stratified Sampling:
    • Used when the population is heterogeneous regarding a confounding variable (e.g., gender).
    • The population is divided into groups (strata) based on the confounding variable.
    • Samples are selected from each stratum proportionally.
    • Example Calculation: If a population of 18 has 9 females (50%50\%) and 9 males (50%50\%), and a sample of 4 is needed: select 2 females and 2 males (0.50×4=20.50 \times 4 = 2 per group).
    • This ensures at least one individual from every subgroup is represented.
  • Cluster Sampling:
    • Used for convenience rather than controlling confounding variables.
    • The population is split into clusters. A random sample of clusters is chosen, and all individuals within the selected clusters are included in the sample.
  • Systematic Sampling:
    • Requires a random starting point and a sampling interval denoted as kk.
    • Calculating kk: k=Population Size (N)Desired Sample Size (n)k = \frac{\text{Population Size (N)}}{\text{Desired Sample Size (n)}}.
    • If kk is not an integer, it is rounded up. For N=18N=18 and n=4n=4, 18/4=4.518/4 = 4.5, so k=5k=5.
    • Starting from the random entry (e.g., entry 2), select every 5th5^{th} individual (2,7,12,172, 7, 12, 17).
    • Drawback: If the population list has an underlying pattern (e.g., alternating genders) and the interval matches that pattern, the sample may be biased.

Sampling Biases

  • Non-response Bias: Occurs when a significant portion of the surveyed group chooses not to respond. This often leaves only those with extreme opinions in the dataset.
  • Sampling Bias: Occurs when certain members of the population are systematically excluded from the sampling process, making the sample unrepresentative.
  • Response Bias: Occurs when the phrasing of questions or the environment influences participants to answer differently than they truly feel (e.g., leading questions like "How much do you appreciate our innovative method?").

Study Designs

  • Observational Study: Researchers observe, watch, or record characteristics and behaviors without intervention.
    • Case Control Study: Identify a "case" group with a specific outcome and a "control" group without it. Researchers look back at past habits (the explanatory variable) to assess associations.
    • Sample Survey: Gathers information from a sample to generalize findings to the whole population.
  • Experimental Study: Researchers impose treatments on subjects to observe specific outcomes.
    • Explanatory Variable (Factor): The variable being manipulated to see its effect (e.g., type of training program).
    • Response Variable: The outcome measured (e.g., academic performance or memory test scores).

Elements of a Good Experiment

  • Control: Using comparison to account for outside factors (e.g., comparing Program A to Program B). This ensures improvements are due to the treatment, not time of day or temperature.
  • Randomization: Randomly allocating subjects to treatment groups to balance confounding variables (like age) that are not accounted for in the design.
  • Replication: Using a large enough sample size in each group to ensure results are not due to individual idiosyncratic traits.

Specific Experimental Designs

  • Completely Randomized Design: Subjects are randomly assigned directly to treatment groups (e.g., 60 people split into 30 for Program A and 30 for Program B).
  • Block Randomized Design: Subjects are first sorted into "blocks" based on a confounding variable (e.g., sorting 60 students by their WAM into High, Medium, and Low groups). Random allocation to treatments is then performed within each block.
  • Matched Pairs Design: A specific type of block design where subjects are paired based on similar characteristics (age, gender). Within each pair of two, one is randomly assigned to treatment A and the other to treatment B.
  • Repeated Measure Design: Each subject receives both treatments (e.g., 30 people get A then B, 30 get B then A).
    • Pros: Useful for small samples as it increases the number of observations (60 people=120 observations60\,\text{people} = 120\,\text{observations}).
    • Cons: Not suitable if the first treatment affects the second (e.g., fatigue or familiarity with test questions).
  • Factorial Experimental Design: Used to test the effects of two or more explanatory variables (factors) simultaneously.
    • Example: Testing Program (A vs. B) and Time of Day (Morning vs. Evening). This creates a 2×22 \times 2 table with 4 treatment combinations.
    • Pros: Cost and time-effective because it studies two factors without needing two separate experiments.

Questions & Discussion

  • Calculators: During exams, students might find they need to remove calculator lids and place them on the floor to avoid suspicion of cheating. Stickers should be placed on the body of the calculator.
  • Random Seed: In Excel, the random seed determines the specific sequence of how numbers are generated. It does not matter what number is used, but using the same seed allows for the replication of the same "random" sequence if needed.
  • Study Materials: Seminar recordings are available roughly 30 minutes after the live session. Pre-recorded theory videos should be watched before attending the seminar.
  • Check-in Quizzes: These are open-book assessments. The first check-in quiz opens at 5:30 p.m.5:30\,\text{p.m.} on the day of the first seminar and closes Sunday at midnight.