Sampling Distribution and Inference - Quick Notes

Sampling distribution basics

• Sampling distribution: distribution of a statistic across all possible samples from a population.
• Key idea: use the sampling distribution to make inferences about a population parameter (mu, p, etc.) from a sample statistic.
• Population vs. sample concepts:
  • Population measures are parameters (e.g., population mean μ, true proportion p).
  • Sample measures are statistics (e.g., sample mean X̄, sample proportion p̂).

Central Limit Theorem (CLT)

• If a random sample is drawn from any population, the sampling distribution of the sample mean X̄ is approximately normal if the sample size is large enough (commonly n ≥ 30).
• If the population is already normal, smaller samples can still yield a normal X̄.
• CLT enables inference using the standard normal distribution after standardization.

Sampling distribution of the sample mean (X̄)

• Mean of the sampling distribution: $\mathbb{E}[\bar{X}] = \mu.$
• Variance of the sampling distribution: $\operatorname{Var}(\bar{X}) = \frac{\sigma^2}{n}.$
• Standard error of the mean: $\text{SE}(\bar{X}) = \frac{\sigma}{\sqrt{n}}.$
• Standardization to Z: $Z = \frac{\bar{X} - \mu}{\sigma/\sqrt{n}}.$
• Finite population correction (when population size N is finite):
  $\sigma_{\bar{X}} = \sigma \sqrt{\frac{N-n}{N-1}} \cdot \frac{1}{\sqrt{n}} = \frac{\sigma}{\sqrt{n}} \sqrt{\frac{N-n}{N-1}}.$
  Note: often ignored when N is large relative to n.

Inference with X̄

• Because X̄ ≈ Normal(μ, σ^2/n) for large n, we can make confidence statements about μ using the standard normal distribution.
• Increasing n reduces the SE and tightens the distribution around μ.

Sampling distribution of the sample proportion p̂

• For Bernoulli trials with true population proportion p, let X be the number of successes in n trials: X ~ Bin(n, p).
• Sample proportion: $\hat{p} = \frac{X}{n}.$
• Mean and variance of p̂:
  $\mathbb{E}[\hat{p}] = p, \quad \operatorname{Var}(\hat{p}) = \frac{p(1-p)}{n}.$
• Standardization: $Z = \frac{\hat{p} - p}{\sqrt{p(1-p)/n}}.$
• Normal approximation to p̂ is valid if the rule of thumb is met: $np \ge 10 \quad\text{and}\quad n(1-p) \ge 10.$
• For X (number of successes), X ~ Bin(n, p) with mean $\mathbb{E}[X] = np$ and variance $\operatorname{Var}(X) = np(1-p).$

Bernoulli trials, independence, and sampling with replacement

• Bernoulli trial: one trial with two outcomes (success/failure).
• Trials must be independent and identically distributed; use sampling with replacement to preserve constant p across trials.

Examples (illustrative calculations)

• Chocolate weights example
  • Given single bar: X ~ Normal(μ=32.2, σ=0.3).
  • Q1: P(X > 32) = ?
  • Z = (32 - 32.2)/0.3 = -0.667 ➜ P(X > 32) ≈ 0.75.
  • Q2: For a pack of n = 4, X̄ ~ Normal(μ=32.2, σ^2/n = 0.3^2/4 = 0.0225).
  • SE = 0.3/√4 = 0.15; Z = (32 - 32.2)/0.15 = -1.333 ➜ P(X̄ > 32) ≈ 0.91.
  • intuition: larger samples pull the average toward the true mean and reduce variability.
• Weekly income example
  • Population: μ = 600, σ = 100, n = 25.
  • P(X̄ < 550) = P(Z < (550-600)/(100/√25)) = P(Z < -2.5) ≈ 0.0062 (0.62%).
• Proportions example (Laurier brand)
  • Population proportion: p = 0.30, n = 1000.
  • Q: P(p̂ > 0.32) = ?
  • Z = (0.32 - 0.30)/√(p(1-p)/n) = 0.02 / √(0.21/1000) ≈ 1.38
  • P(p̂ > 0.32) ≈ 1 - Φ(1.38) ≈ 0.08 (8%).

Finite population correction (recap)

• When sampling from a finite population, the standard error is adjusted by: $\text{SE} = \frac{\sigma}{\sqrt{n}} \sqrt{\frac{N-n}{N-1}}.$
• For very large N, the correction factor ~ 1 and is often ignored.

Quick takeaways

• The sampling distribution describes how a statistic would behave over many samples, enabling inference about population parameters.
• X̄ has mean μ and variance σ^2/n; SE shrinks as n grows; CLT guarantees approximate normality for large n.
• p̂ has mean p and variance p(1-p)/n; normal approximation requires np ≥ 10 and n(1-p) ≥ 10.
• Standardize to Z to use the standard normal table for probabilities.
• Larger samples reduce variability and improve accuracy for population parameters; hypotheses and confidence statements rely on these distributions.
• Always consider whether finite population correction is needed when N is not much larger than n.

Quick formulas to memorize

• For X̄: $\mathbb{E}[\bar{X}] = \mu,\quad \operatorname{Var}(\bar{X}) = \frac{\sigma^2}{n},\quad \text{SE}(\bar{X}) = \frac{\sigma}{\sqrt{n}}.$
• Standardize X̄: $Z = \frac{\bar{X}-\mu}{\sigma/\sqrt{n}}.$
• For p̂: $\mathbb{E}[\hat{p}] = p,\quad \operatorname{Var}(\hat{p}) = \frac{p(1-p)}{n},\quad Z = \frac{\hat{p}-p}{\sqrt{p(1-p)/n}}.$
• Normal approximation conditions for p̂: $np \ge 10,\; n(1-p) \ge 10.$
• Finite population correction (FPC): $\text{SE}_{\bar{X}} = \frac{\sigma}{\sqrt{n}} \sqrt{\frac{N-n}{N-1}}.$