Parameters, Estimators & Sampling Distributions

Objectives & Learning Outcomes

• By the end of the lesson you should be able to:
  • Define & distinguish parameters vs. estimators ("statistics").
  • Recognize random variables & interpret their probability distributions (tables, graphs, PDFs).
  • Explain sampling distributions, the factors that control their shape, centre & spread, and why they matter for inference.
  • Describe—in detail—the sampling distribution of the sample mean from simple random samples (with replacement).
  • Connect these ideas to inference tools planned in the PPDAC framework.

Parameters

• Definition: a single number that summarizes a characteristic of the entire population ⇒ constant for that population.
• Common parameter types
  • Typical‐value: mean, median, mode.
  • Frequency: proportion/percentage.
  • Extremes: maximum, minimum.
  • Variability: variance, standard deviation ($\sigma$), range.
  • Relationship: regression slope & intercept.
• Example (high‐heel study)
  • Response variable: perceived attractiveness.
  • Parameter of interest: difference in population mean attractiveness between women wearing high vs. flat heels.

Estimators (Statistics) & Sampling Variability

• Statistic / Estimator: number that summarizes a characteristic for a sample → random because samples differ.
• Sampling variability: sample‐to‐sample fluctuation in a statistic.
  • Illustrated by repeatedly sampling $n = 16$ IQ scores; observed variation in $\bar x$.
• Terminology link
  • Parameter = population constant (e.g.
    $\mu$).
  • Statistic = sample value (e.g.
    $\bar x$).

Statistical Inference

• Goal: draw conclusions about population parameters using only sample data.
• Challenge: every statistic contains sampling error (random deviation from parameter).
• Inference procedures (null‐hypothesis tests, confidence intervals, model fitting) use knowledge of sampling distributions to quantify that error.
• These tools are planned during Plan step of PPDAC.

Factors Influencing Estimators

1.
Sampling Error

• Difference parameter − statistic for a good (representative) sample.
• Magnitude ↓ as sample size $n$ ↑.
  2.
  Bias
• Systematic favouring of certain outcomes/individuals while sampling ⇒ pushes statistics consistently away from parameter.
• Must be minimized by sound design.

Random Variables

• Definition: numerical variable whose value is generated by a random process.
• Notation
  • Upper-case (e.g.
    $X,Y$) ⇒ variable; lower-case + subscripts $x<em>1,x</em>2,\ldots$ ⇒ particular outcomes.
• Examples
  • $X$ = number of days (0–7) a random adult exercised ≥1 h last week (discrete).
  • $Y$ = hours a teenager spent on social media on a randomly selected day (0–24) (continuous).

Probability Distributions

• Tell us which values a random variable can take and how likely each is.
• Representations: equations, tables, graphs.
• Must list all values and have total probability $=1$.
• Table example: $P(X=3)=0.32$ implies a 32 % chance the adult exercised on 3 days.

Continuous Variables & Density Curves

• Infinite possible values ⇒ $P(X=k)=0$ for any exact point.
• Use density curve: non-negative, total area $=1$.
  • Area over interval = probability.
• Illustration: nitrate concentration $Y$ in water, domain 0–1 mg/L.
  • $P(0.6\le Y \le 0.8)=0.42$ (= shaded area).

Sampling Distributions

• Sampling distribution: probability distribution of a sample statistic over all possible samples of size $n$.
• Consequently statistics themselves are random variables.

Recognizing Sampling Distributions (Quiz Results)

• Correct examples
  • 10 separate samples of 14 ducks; histogram of $\bar X$ ⇒ approximate sampling distribution.
  • Program enumerates every $n=25$ sample & plots minima ⇒ exact sampling distribution.
• Incorrect
  • Single SRS of 60 crickets; histogram of raw lengths = just a sample, not a sampling distribution.

Sampling Distribution of the Mean

Simulation Findings

• Populations examined via applet
  1.
  Normal: $\mu=16,\;\sigma=5$.
  2.
  Right-skew: $\mu=8.08,\;\sigma=6.22$.
• Observed patterns
  • Centre: $\mu_{\bar X}\approx\mu$ for any $n$.
  • Spread: $\sigma_{\bar X}<\sigma$ and ↓ as $n$ ↑.
  • Shape: Normal population ⇒ sampling distribution Normal for all $n$; non-Normal population ⇒ skew for small $n$ but approaches Normal when $n$ large.

Formal Properties

Let $X$ be the population variable, $\bar X$ the sample mean from an SRS of size $n$.

• Mean (expected value)
  $\mu_{\bar X}=\mu$
• Standard error (spread)
  $\sigma_{\bar X}=\frac{\sigma}{\sqrt{n}}$
• Shape rules
  1.
  If population Normal ⇒ $\bar X$ exactly Normal.
  2.
  If population not Normal ⇒ $\bar X$ approximately Normal when $n$ “large.”

Central Limit Theorem (CLT)

• States that for sufficiently large $n$, distribution of $\bar X$ tends toward $\text{Normal}(\mu,\sigma^2/n)$ regardless of population shape.
• Practical threshold (conservative): $n\ge50$.
• Enables use of Normal probability methods for inference on means.

Practical & Ethical Implications

• Sampling design (SRS, stratified, cluster) influences sampling distribution properties.
• Larger $n$ reduces sampling error but costs resources; planning balances precision & feasibility.
• Avoiding bias upholds research integrity and validity of conclusions.

Key Formulae & Symbols

• Population mean: $\mu$
• Population SD: $\sigma$
• Sample mean (statistic): $\bar x$ (realization), $\bar X$ (random variable)
• Standard error of $\bar X$: $\sigma_{\bar X}=\sigma/\sqrt{n}$
• CLT approximation: $\bar X \approx \text{Normal}(\mu,\,\sigma^2/n)$ for large $n$.

Connections & Applications

• Foundations for forthcoming topics: confidence intervals, t-tests, regression.
• Real-world examples: estimating mean water contaminants, polling enjoyment of an online course, comparing attractiveness ratings.
• Reinforces Plan stage of PPDAC: select analyses aligned with sampling distribution properties.