PSTAT 5LS – Theory-Based Inference for a Population Proportion

Course Logistics and Housekeeping

• Course: PSTAT 5LS – Theory-Based Inference for $p$ (Slide Set 5)
• Today’s topic: Introduction to theory-based inference for a population proportion $p$
• Next time: Continuation of the same topic
• Upcoming homework deadlines
  • HW 2 due Tue Jul 8 @ 11:59 PM
  • HW 3 due Mon Jul 14 @ 11:59 PM
  • (HW 4 appears on slides but due date not shown: “F <118”)
• Office-hours reminder
  • Instructor OH: Tue & Thu 2–3 PM via Zoom
  • Encouragement: “Visit us in office hours!”

Bridging Simulation & Theory

• Previous slide set used simulation (randomization & bootstrap) to approximate sampling distributions.
• Empirical observation: these simulated sampling distributions looked nearly normal.
  • Example 1 (dolphin communication): histogram of simulated proportion of correct guesses.
  • Example 2 (community recycling): histogram of simulated proportion of recyclers.
• Take-away: the normal pattern hints that theory (specifically the Central Limit Theorem) can describe $\hat p$’s behaviour without repeated simulation.

Sampling Distributions

• Definition: A sampling distribution is the distribution of a statistic (e.g., sample proportion $\hat p$ or sample mean $\bar x$) over all possible random samples of fixed size $n$ from the population.
  • Describes shape, centre, and variability attributable purely to random sampling.
  • Knowing this distribution lets us judge how “unusual” any observed statistic is when $H_0$ is true.

Distribution of the Sample Proportion $\hat p$

• Mean (centre): $E(\hat p)=p$ (the true population proportion).
• Standard deviation (variability), termed standard error (SE):
  $SE = \sqrt{ \dfrac{p(1-p)}{n} }$
  • Acts as a new “ruler” to quantify how far an observed $\hat p$ is from the hypothesised mean.

Central Limit Theorem (CLT) for Proportions

• When conditions are met, the sampling distribution of $\hat p$ is approximately normal:
  $\hat p \sim N\Bigl(\, p,\,\dfrac{p(1-p)}{n} \Bigr).$
• Importance: Allows analytic (theory-based) inference instead of computational simulation.

Conditions Required for Normal Approximation

1. Independence (a.k.a. Randomness) Condition
   • Individual observations must not influence each other.
   • Usually satisfied by a simple random sample (SRS) or well-designed randomised study.
   • If sampling without replacement, ensure population is at least 10× larger than sample.
2. Success–Failure Condition
   • Expected counts—not necessarily observed counts—must include at least 10 successes and 10 failures.
   • For inference on $p$ we check:
   $np \ge 10 \quad\text{and}\quad n(1-p) \ge 10.$
   • For hypothesis tests we substitute $p<em>0$ (value posited by $H</em>0$) because under $H<em>0$ we assume $p=p</em>0$.
   • The threshold “10” is chosen empirically: ensures normal curve approximates the true distribution well enough.

Using the Normal Model in Hypothesis Testing

• Unknown true $p$ → plug in hypothesised $p<em>0$ when computing SE during a test. $SE</em>{H<em>0} = \sqrt{ \dfrac{p</em>0(1-p_0)}{n} }$
• This substitution keeps calculations self-consistent with the null model.

Formal Steps for a Proportion Hypothesis Test

1. State hypotheses
   • Null: $H<em>0 : p = p</em>0$
   • Alternative: $H<em>A : p < p</em>0,\; p\ne p<em>0, \text{ or } p > p</em>0$ (direction dictated by research question before seeing data).
2. Check conditions (independence + success–failure using $p_0$).
3. Compute test statistic (z-score):
   $z = \dfrac{\hat p - p<em>0}{\sqrt{\dfrac{p</em>0(1-p_0)}{n}}}.$
4. Find p-value
   • Use normal distribution areas; tail(s) chosen according to $H_A$.
5. Decision
   • Compare p-value to significance level $\alpha$ (commonly 0.05).
   • p-value $\le \alpha$ → reject $H<em>0$ (result is “statistically significant”). • p-value $> \alpha$ → fail to reject $H</em>0$ (not significant).
6. Contextual conclusion
   • Translate statistical outcome to plain language relating to the study subject.

Interpreting the z Test Statistic

• $z$ measures distance of observed $\hat p$ from $p_0$ in standard-error units.
  • Example: $z=2$ means observation lies 2 SEs above hypothesised mean.
• Link to Empirical (68-95-99.7) Rule:
  • |z|≈1 → ordinary; |z|≈2 → somewhat unusual; |z|≈3 → very rare under $H_0$.
  • For non-integer z, exact areas require technology.

Computing Areas with R’s pnorm()

• Syntax: pnorm(q, mean = 0, sd = 1, lower.tail = TRUE)
  • q = z-score (quantile).
  • lower.tail = TRUE returns $P(Z \le q)$, FALSE returns P(Z > q).
• Tail selection depends on $H_A$:
  • Left-tailed (

Worked Examples

1. Dolphin Communication (Doris & Buzz)

• Hypotheses:
  H0: p = 0.50 \quad vs. \quad HA: p > 0.50.
• Data: $n=16,\; \hat p = 15/16 = 0.9375.$
• Test statistic:
  $z = \dfrac{0.9375 - 0.50}{\sqrt{\dfrac{0.50\times 0.50}{16}}} = 3.50.$
• p-value (right tail):
  pnorm(3.50, mean = 0, sd = 1, lower.tail = FALSE) → $0.0002326291.$
• Interpretation: Extremely small p-value → strong evidence dolphins / Buzz perform better than chance.

2. Community Recycling Rate

• Hypotheses:
  $H<em>0: p = 0.70 \quad vs. \quad H</em>A: p \ne 0.70.$
• Data: $n=800,\; \hat p = 530/800 = 0.6625.$
• Test statistic:
  $z = \dfrac{0.6625 - 0.70}{\sqrt{\dfrac{0.70\times 0.30}{800}}} = -2.315.$
• p-value (two-tailed):
  2*pnorm(-2.315, mean = 0, sd = 1, lower.tail = TRUE) → $0.020613.$
• Interpretation: At $\alpha=0.05$, result is statistically significant (p ≈ 0.021 < 0.05); observed recycling proportion differs from 70% (specifically, it is lower).

Visual Memory Aids

• Normal curve with mean $p_0$ labelled at centre, tick marks at $\pm SE, \pm2SE, \pm3SE$.
  • Approx. 68% within ±1 SE, 95% within ±2 SE, 99.7% within ±3 SE.
  • Extreme regions (tails) correspond to p-value areas.

Tail Direction Summary

• Alternative $H<em>A$ dictates tail(s) used: • $p < p</em>0$ → left tail.
  • p > p0 → right tail. • $p \ne p</em>0$ → both tails (double smaller area).

Decision Table (α-Level Rule)

• If p-value ≤ $\alpha$ ⇒ Reject $H_0$ ⇒ “statistically significant.”
• If p-value > $\alpha$ ⇒ Fail to reject $H_0$ ⇒ “not statistically significant.”

Ethical & Practical Considerations (implicit reminders)

• Independence assumption critical; violating it invalidates SE & p-values.
• Choosing $H_A$ direction must come before examining data to avoid bias.
• Success–failure rule ensures ethical communication of uncertainty: prevents overstating evidence when normal approx. is unreliable.
• Real-world relevance: findings (e.g., dolphin cognition, community recycling) can inform environmental or cognitive research initiatives.

Connections to Earlier Material

• Builds directly on earlier simulation-based inference: both aim to quantify sampling variability.
• Standardised statistic $z$ is analogue to earlier randomisation-distribution counts of “how far from center” but now parameterised by theoretical SE.