Foundations of Statistical Inference

Course Orientation & Ultimate Goal

• Statistics viewed as both science and art for working with data.
• Fundamental mission:
  • Collect, organize, summarize, describe, and draw inferences from data.
  • Build a conceptual foundation before diving into computational methods.
• Guiding question: How do we use information from a sample to learn about a population?

Key Definitions

• Statistics (discipline)
  • Complete toolkit for handling data life-cycle: collection → organization → summarization → analysis → decision-making.
• Data / Datum
  • Plural "data" (singular "datum"); any pieces of information, not necessarily numerical.
• Population
  • “Big group” of interest; could be people, objects, events, voters, etc.
  • Often very large (size denoted $N$, typically unknown).
• Sample
  • Subset of the population actually observed (size denoted $n$, always known).
  • Goal: sample should represent the population to ensure valid inference.
• Parameter
  • Fixed (constant) numerical summary of a population (e.g., true mean, true proportion).
  • Unknown and unchanging for the population.
• Statistic
  • Numerical summary calculated from a sample (e.g., sample mean $\bar x$, sample proportion $\hat p$).
  • Random/variable because it changes from sample to sample.
  • Mnemonic: Parameter ↔ Population; Statistic ↔ Sample.
• Variable
  • Measured characteristic that can vary across subjects (e.g., height, vote choice).
• Constant
  • Quantity that does not vary (e.g., a specific parameter value, a universal physical constant).

Illustrative Study: Toxin & Time-to-Conception

• Research hypothesis: Women exposed to a certain toxin require longer to conceive.
• Data collected: “Time needed to conceive” for women with differing toxin exposures.
• Observed result: Greater exposure → longer conception time.
• Cautionary tale:
  • Possible confounder: mothers’ smoking habits during pregnancy may influence daughters’ fertility.
  • If unmeasured, we may falsely attribute longer conception time to toxin instead of prenatal smoke exposure.
• Historical parallel: 1980s study linking coffee to heart disease later found the observed association was actually due to smoking prevalence among coffee drinkers.
• Moral: Failure to control external factors leads to wrong decisions even when statistical procedures are applied correctly to the measured variables.

Uncertainty & Probability

• Inference based on a subset introduces uncertainty ⇒ decisions can be right or wrong.
• Need to quantify uncertainty via probability.
  • Probability measures the likelihood of outcomes; values fall in $[0,1]$.
  • $0$ ⇒ impossible; $1$ ⇒ certain.
  • Probability statements apply only to variables, not constants (parameters).

Visual Framework (Textbook Diagram)

• Population (unknown “reality”) → sampling (size $n$) → compute sample summaries/histogram → infer about a population model (e.g., Normal, Binomial, Poisson) → attach probability-based certainty measure.
• Assumptions about underlying distribution critical for valid inference.

Two Branches of Statistics

• Descriptive Statistics
  • Collection, organization, visualization (charts, tables), and summarization (mean, median, SD) of data.
  • Core of high-school level statistics.
• Inferential Statistics
  • Uses sample statistics to draw conclusions about population parameters and quantify uncertainty.
  • Driving force of modern scientific research.
• Course plan: Master descriptive tools first, then develop inferential methods.

Notation Recap

• Population size: $N$ (often large, unknown).
• Sample size: $n$ (known, chosen by researcher).

Variable Taxonomy

• By measurement scale:
  • Quantitative (numeric values make arithmetic sense).
  • Qualitative / Categorical (labels, categories, levels).
• By numerical nature (for quantitative variables):
  1. Discrete Random Variable
  • Takes countable, separated values (gaps exist).
  • Example: number of people in line; ring sizes (9, 9.5, 10…).
  1. Continuous Random Variable
  • In theory can take any real value within an interval (no gaps).
  • Example: precise finger circumference in millimetres; human height.
• Random Variable
  • Numerical outcome whose exact value is unknown until observation.
  • If qualitative, we often convert to numbers via counts (still yields a discrete random variable).

Examples & Clarifications

• Ring size (standard jeweler increments) ⇒ discrete.
• Measuring pinky circumference with a tape measure ⇒ continuous (any fraction of a millimetre possible in theory).
• Rolling a fair die: outcome before roll is a discrete random variable with values ${1,2,3,4,5,6}$.

Concept Checks

• Parameter: constant (no variability).
• Statistic: variable (changes with each sample).
• Probability only meaningful for variables.

Reading & Next Steps

• Review textbook Section 1 for reinforcement; attempt "Check-Your-Understanding" questions (solutions in appendix).
• Independently read Sections 2 & 3 (no accompanying videos).
• Next lecture/video will begin with Section 4.