Theory-Based Inference for a Population Proportion (p)

Introduction & Course Logistics

• Welcome to PSTAT 5LS – Slide Set 5: Theory-Based Inference for p
  • Transitioning from simulation-based inference to theory-based (normal-model) inference.
• Administrative reminders (Summer session examples*)
  • HW 2 due Tue July 8 • 11:59 PM
  • HW 3 due Mon July 14 • 11:59 PM
  • Office hours (instructor example): T & R 2–3 PM, via Zoom

From Simulation to Theory

• Earlier slide set: simulated many samples under $H_0$ and plotted the resulting sample proportions.
  • Dolphin–communication example and community-recycling example both showed bell-shaped histograms.
• Observation → Insight
  • Repeated-sample behavior of $\hat p$ resembles a normal curve → motivates normal approximation.
  • A theoretical model lets us replace thousands of shuffles with a single formula-based calculation.

Sampling Distribution Fundamentals

• Sampling distribution = distribution of a statistic across all possible samples of fixed size n.
  • Describes shape, center, spread attributable to random sampling ("chance alone").
• Key takeaway: If we know this distribution, we can quantify how unusual a single sample’s statistic is when $H_0$ is true.

Distribution of the Sample Proportion $\hat p$

• Center (mean): $E[\hat p]=p$ (the true population proportion).
• Spread (standard error SE): $SE_{\hat p}=\sqrt{\dfrac{p(1-p)}{n}}$
  • Think of SE as a new ruler for measuring how far an observed $\hat p$ sits from the hypothesised $p$.

Central Limit Theorem (CLT) for Proportions

• CLT assures that, under certain conditions, the sampling distribution of $\hat p$ is approximately normal:
  $\hat p \sim N\Big(p,\; \dfrac{p(1-p)}{n}\Big)$
• Implication: We can use z-scores & normal probabilities for inference instead of re-simulating.

Conditions for Normal Approximation

1. Independence
   • Individual observations must not influence one another.
   • Usually guaranteed by simple random sampling (SRS) or a well-designed experiment.
2. Success–Failure Condition
   • Under the model being checked (often $H<em>0$), require $np</em>0\ge 10 \quad\text{and}\quad n(1-p_0)\ge 10$
   • Ensures tails of the binomial are well-captured by the normal curve.
   • "10" is empirical but widely accepted; larger thresholds tighten approximation.

Using the Normal Model in Hypothesis Testing

• In practice we don’t know $p$, so for SE we plug in the null value $p<em>0$: $SE</em>{H<em>0}=\sqrt{\dfrac{p</em>0(1-p_0)}{n}}$
• Interpretation: We build the sampling distribution assuming $H_0$ is true; deviations are judged relative to that benchmark.

Independence: Deeper Considerations

• Random sampling → independence usually reasonable.
• Non-random scenarios (e.g.
  cluster sampling, time series, social networks) → must justify or adjust (e.g.
  finite-population correction, bootstrapping, or other models).

Success–Failure Revisited for Tests vs CIs

• Hypothesis test: check $np<em>0,\; n(1-p</em>0)$ (expected counts) ≥ 10.
• Confidence interval: replace $p_0$ with observed $\hat p$.

Six Classic Steps in a One-Proportion z-Test

1. State hypotheses $H0: p=p0, choose $H_A$ (
2. Check conditions (independence, success-failure).
3. Compute test statistic
   $z=\dfrac{\hat p - p<em>0}{\sqrt{p</em>0(1-p_0)/n}}$
4. Find p-value from the standard normal distribution.
5. Decision: compare p-value to $\alpha$.
6. Contextual conclusion in plain language.

Formulating Hypotheses Correctly

• $p_0$ comes from the research question, not the data (e.g.
  historical rate, theoretical expectation, regulatory threshold).
• Possible alternatives:
  • Left-tailed: $H<em>A: p < p</em>0$ (looking for a decrease).
  • Right-tailed: $H<em>A: p > p</em>0$ (looking for an increase).
  • Two-tailed: $H<em>A: p \neq p</em>0$ (any change).

Standardization & the 68-95-99.7 Rule Refresher

• Converting to a z-score (test statistic) expresses deviation in SE units.
  • $|z|=1 \Rightarrow$ (~68\%) of null sample proportions lie closer than you.
  • $|z|=2 \Rightarrow$ (~95\%) lie closer.
  • $|z|=3 \Rightarrow$ (~99.7\%) lie closer.
• For non-integer |z| (e.g.
  1.75) need software for exact areas.

Computing p-Values with R’s pnorm()

• Generic syntax: pnorm(q, mean=0, sd=1, lower.tail=TRUE)
  • q = z-score (quantile).
  • lower.tail=TRUE → $P(Z \le q)$; FALSE → $P(Z \ge q)$.
• Tail selection depends on $H_A$:
  • Left-tailed (
  • Right-tailed (>): p-value = pnorm(z, lower.tail=FALSE).
  • Two-tailed (≠): p-value = 2*pnorm(|z|, lower.tail=FALSE).

Worked Example 1 – Dolphin Communication

• Data: Buzz guessed 15/16 correct → $\hat p=0.9375$.
• Hypotheses: $H<em>0:p=0.50 \quad vs \quad H</em>A:p>0.50$ (right-tailed).
• Test statistic:
  $z=\dfrac{0.9375-0.50}{\sqrt{0.50(1-0.50)/16}}=3.50$
• p-value (R):

  pnorm(3.50, lower.tail = FALSE)  # 0.0002326

• Conclusion (e.g. with $\alpha=0.05$): Reject $H_0$ → convincing evidence dolphins communicate.

Worked Example 2 – Community Recycling

• Data: 530/800 households recycle → $\hat p=0.6625$.
• Hypotheses: $H<em>0:p=0.70 \quad vs \quad H</em>A:p\neq0.70$ (two-tailed).
• Test statistic:
  $z=\dfrac{0.6625-0.70}{\sqrt{0.70(1-0.70)/800}}=-2.315$
• p-value (R):

  2*pnorm(-2.315)  # 0.0206

• With $\alpha=0.05$: Reject $H_0$ → recycling rate differs from 70 %.

Tail Direction & p-Value Recap

• p-value = probability of results as or more extreme in the direction of $H_A$.
  • Make sure to multiply by 2 only for two-tailed tests.

Decision Rules & Statistical Significance

• If $p\le\alpha$ → Reject $H_0$ (statistically significant).
• If $p>\alpha$ → Fail to reject $H_0$ (not enough evidence).
• Common $\alpha$ levels: 0.10, 0.05, 0.01 (context-driven; e.g.
  medical trials often demand 0.01).

Practical & Ethical Considerations

• Adequate sample size → meets success-failure & yields useful power.
• Independence violations (e.g.
  sampling classmates, social media followers) bias SE and inflate Type-I error.
• Overreliance on p-values: Always accompany with effect size and confidence interval.
• Ethical reporting → state assumptions, limitations, and potential multiple-testing issues.

Connections & Next Steps

• Link to simulation: Theory approximates what we previously estimated via randomization.
• Upcoming lectures: theory-based confidence intervals for $p$, two-sample proportion tests, and extensions when conditions fail (e.g.
  exact binomial tests, bootstrap CIs).