Comprehensive Notes on Sampling, Sampling Distributions & Statistical Inference

Data & Statistics Overview

• Data: numerical or categorical measurements collected in a study.

• Goal of statistics: organize and interpret data to extract meaningful information.

• Two broad branches:

  • Descriptive statistics → organize & summarize data already in hand.

  • Inferential statistics → generalize from a sample to a larger population, acknowledging uncertainty.

Descriptive Statistics

• Provide concise summaries of data.

  • Tables, frequency distributions, graphs (histograms, bar charts, boxplots).

  • Summary numbers (measures of center, spread, shape, position).

• Terminology:

  • Parameter: descriptive value computed for an entire population.

  • Statistic: descriptive value computed for a sample and used to estimate a parameter.

Inferential Statistics

• Use sample information to draw conclusions (inferences) about the population.

• Recognize that sample statistics are generally imperfect proxies for population parameters because only a subset is observed.

• Central activity: quantify and control the error introduced by sampling.

Sampling Basics

• Population: totality of elements we wish to study (e.g.
  1,000 college students).

• Sample: subset chosen from the population (e.g.
  5 selected students).

• Census: complete enumeration of every population element (time-consuming, costly, sometimes destructive).

• Survey: data-collection procedure performed on a sample.

• Sampling error: $\text{Sampling error}=\text{Statistic}-\text{Parameter}$ , an inevitable discrepancy due to observing only part of the population.

• Systematic error (bias): consistent, directional error; arises from flawed study design or implementation (e.g.
  non-coverage, measurement bias).

Key Terms & Concepts

• Bias: occurs when certain groups are systematically excluded or over-represented so that the sample is not representative.

• Target population: group to which researchers intend to generalize their findings.

• Sampling unit: smallest unit eligible for selection (person, household, bottle, etc.).

• Sampling frame: operational list that enumerates every sampling unit.

• Sampling scheme: explicit rule specifying how units are drawn from the frame.

• Fundamental rule of inference: We may generalize from available data only if the sample is representative with respect to the research question.

Why Sampling?

• Economical relative to a census (lower cost, less field time).

• Feasible when studying the whole population is impossible.

• Prevents destruction of units when measurement is destructive (e.g.
  crash-testing vehicles).

Types of Sampling

• A good sample is unbiased and representative.

• Probability sampling: every population element has a known (not necessarily equal) probability of selection.

  • Examples: Simple Random Sampling (SRS), stratified, cluster, systematic, multistage.

• Non-probability sampling: selection probabilities unknown (convenience, judgement, quota); threatens validity of inference.

Simple Random Sampling (SRS)

• Each possible sample of size $n$ has the same probability of being chosen.

• Provides tractable mathematics for sampling distributions.

Sampling Error

Systematic Error

• Sampling error (random error): variability that arises purely by chance; summarized by the standard error (SE).

  • Example SE for mean: $\sigma_{\bar{x}}=\frac{\sigma}{\sqrt{n}}$.

• Systematic error (bias): errors that shift results in one direction.

  • Selection bias, information (measurement) bias, nonresponse bias, etc.

Point Estimation & Margin of Error

• Point estimate: single-number best guess of a parameter (e.g.
  $\bar{x}=64$ rating points).

• Margin of error (MoE): $\text{MoE}=z<em>{\alpha/2}\,\sigma</em>{\bar{x}}$, creates an interval $\bar{x}\pm \text{MoE}$ capturing the parameter with specified confidence (e.g.
  $\pm 3\%$ around a presidential approval rating).

• Directly derived from the sampling distribution.

Sampling Distributions

• Because statistics vary from sample to sample, each statistic is a random variable with its own probability distribution — the sampling distribution.

• Common statistics & their distributions under SRS:

  • Sample mean $\bar{x}$.

  • Sample variance $s^2$.

  • Sample proportion $\hat{p}$.

• Example (M&M packs): means of all possible size-2 samples (4–10 yellow pieces) take values 4.5, 5, 5.5, …, 9.

  • Each mean has probability $0.1$; the set forms the sampling distribution of $\bar{x}$.

  • One can compute its expected value $E(\bar{x})$ and variance $Var(\bar{x})$.

Central Limit Theorem (CLT)

• Statement: For simple random samples of size $n$ from a population with mean $\mu$ and variance $\sigma^2$, the distribution of $\bar{x}$ approaches normality as $n$ becomes large (rule of thumb: $n\ge 30$).

  • Mean of $\bar{x}$: $E(\bar{x})=\mu$.

  • Variance: $Var(\bar{x})=\sigma^2/n$.

• Practical implication: allows normal-based inference even when the underlying population is non-normal, provided the sample is large.

• If n<30, normality of $\bar{x}$ requires that the population itself be normal.

Standard Error of the Mean

• Infinite (or very large) population: $\sigma_{\bar{x}}=\frac{\sigma}{\sqrt{n}}$.

• Finite population of size $N$ without replacement: $\sigma_{\bar{x}}=\frac{\sigma}{\sqrt{n}}\sqrt{\frac{N-n}{N-1}}$ (finite-population correction, FPC).

Worked Examples

Example 1: SWS Presidential Rating

• Population rating mean unknown; sample of $n=1200$ respondents gave $\bar{x}=64$, $s=3$.

• Mean of sampling distribution: $E(\bar{x})=64$.

• Standard error (assuming large population): $\sigma_{\bar{x}}=\frac{3}{\sqrt{1200}}\approx0.0866$.

• 95% MoE: $1.96\times0.0866\approx0.17$ rating points ≈ $\pm3\%$ when re-expressed.

• Interpretation: true population rating likely between 61 – 67 (illustrated graphically).

Example 2: Diagnostic Test (30 Students)

• Population: $\mu=75$, $\sigma=5$.

• Sample size $n=30$.

• $E(\bar{x})=75$.

• $\sigma_{\bar{x}}=\frac{5}{\sqrt{30}}\approx0.9129$.

• Probability P(\bar{x}<74):

  • $z=\dfrac{74-75}{0.9129}\approx-1.10$.

  • P(Z<-1.10)=0.1357 ⇒ 13.57% chance the sample mean falls below 74.

Example 3: Juice Bottling Machine

• Calibrated target: $\mu=600\,\text{mL}$; $\sigma=10\,\text{mL}$; $n=36$.

• $E(\bar{x})=600$.

• $\sigma_{\bar{x}}=\dfrac{10}{\sqrt{36}}=1.667$.

• (B) Probability \bar{x}<598:

  • $z=\dfrac{598-600}{1.667}\approx-1.20$.

  • P(Z<-1.20)=0.1151 ⇒ 11.51%.

• (C) Probability \bar{x}>601:

  • $z=+0.60$; P(Z>0.60)=0.2743 ⇒ 27.43%.

• (D) Probability 595<\bar{x}<605:

  • $z<em>{lower}=-3.00$ (0.0013), $z</em>{upper}=+3.00$ (0.9987).

  • Probability $=0.9987-0.0013=0.9974$ → 99.74%.

Example 4: Diagnostic Test (25 Students)

• Population: $\mu=60$, $\sigma=4$, $n=25$.

• (A) $E(\bar{x})=60$, $\sigma_{\bar{x}}=\frac{4}{\sqrt{25}}=0.8$.

• (B) P(\bar{x}<59):

  • $z=\dfrac{59-60}{0.8}=-1.25$.

  • P(Z<-1.25)=0.1056 ⇒ roughly 10.56%.

Illustrative Scenario: M&M Packs

• 5 packs contain {4, 5, 6, 8, 10} yellow pieces.

• All possible size-2 samples (10 possible combinations, equally likely $0.1$ each).

• Sample means range 4.5 – 9, demonstrating variability in $\bar{x}$.

• One can tabulate the sampling distribution, compute $E(\bar{x})$ and $Var(\bar{x})$, reinforcing that $\bar{x}$ is a random variable.

Ethical, Philosophical & Practical Implications

• Representative sampling underpins ethical generalization; biased designs may misinform policy.

• Transparency about sampling error and MoE nurtures public trust.

• Over-reliance on non-probability samples (e.g.
  online polls) can mislead.

• Destructive testing highlights trade-offs between learning and preserving units (e.g.
  product safety vs.
  inventory loss).

Connections & Real-World Relevance

• CLT connects to previous lectures on probability distributions by providing the bridge from any distribution to approximate normality.

• Sampling distributions form the theoretical basis for confidence intervals and hypothesis tests covered later.

• Concepts of bias and representativeness resonate with research design, epidemiologic studies, quality control, and social-science polling.