PSTAT 5LS – Theory-Based Inference for p (Slide Set 5)

Course Logistics and Administrative Details

• Course: PSTAT 5LS (Statistics)
• Current Topic: Theory-Based Inference for a Population Proportion $p$ (Slide Set 5)
• Lecture Timeline
  • Today: Begin Slide Set 5 (Theory-Based Inference for $p$)
  • Next Time: Continue/finish Slide Set 5
• Upcoming Deadlines (Summer session examples)
  • HW 2 due Tue Jul 8, 11:59 PM
  • HW 3 due Mon Jul 14, 11:59 PM
  • HW 4 noted on slides (exact date partially obscured)
• Office Hours
  • Instructor: Tue & Thu, 2–3 PM via Zoom (encouraged to attend)

Transition: From Simulation-Based to Theory-Based Methods

• Previous slide sets used computer‐generated randomization/simulation distributions.
• Empirical finding: these simulated distributions of $\hat p$ looked approximately normal.
  • Example graphics shown
  • “Buzz the dolphin” correct-button guesses ⇒ bell-shaped curve for simulated $\hat p$.
  • Community recycling rate simulation likewise centered & symmetric.
• Insight: The empirical bell shapes hint that a mathematical normal model can replace repeated simulations once its validity is justified.

Sampling-Distribution Fundamentals

• Sampling distribution: distribution of any sample statistic (e.g., $\hat p,\; \bar x$) over all possible samples of fixed size n from the population.
  • Describes shape, center, variability due solely to random sampling.
  • Lets us quantify how “unusual” one observed statistic is when a null hypothesis is true.

Distribution of the Sample Proportion $\hat p$

• Center: $E[\hat p]=p$ (the true population proportion).
• Variability: Standard Error (SE)$\text{SE}=\sqrt{\dfrac{p(1-p)}{n}}$
  • Acts as a new “ruler” for gauging how far an observed $\hat p$ is from $p$.

Central Limit Theorem (CLT) for Proportions

• When certain conditions are met, the sampling distribution of $\hat p$ is approximately normal with
  • Mean $p$
  • Standard deviation $\text{SE}=\sqrt{p(1-p)/n}$.
• Enables theory-based inference (z tests & CIs) rather than simulation.

Conditions Required for the Normal Approximation

1 Independence

• Observations must not influence one another.
  • Usually reasonable if data came from a simple random sample.
  • If sampling without replacement from a finite population of size $N$, check the 10 % rule (sample $n \le 0.10N$) to treat draws as independent.

2 Success–Failure Condition (S–F)

• Need at least 10 expected successes & 10 expected failures.
  • “Success” = category of interest; “failure” = the other category.
• For confidence intervals we use $\hat p$: $n\hat p \ge 10$ and $n(1-\hat p) \ge 10$.
• For hypothesis tests we use the null value $p<em>0$: $np</em>0 \ge 10$ and $n(1-p_0) \ge 10$.
  • Rationale: when $H<em>0$ is true, the true proportion equals $p</em>0$, so expected counts derive from $p_0$, not from data.
• The cutoff “10” is empirical—balances approximation accuracy vs. practicality.

Hypothesis-Testing Framework for a Single Proportion

1. State hypotheses
   • Null: $H<em>0!: p = p</em>0$
   • Alternative (choose
   • $p_0$ is supplied by context, never from data.
2. Check conditions (Independence & S–F as above).
3. Compute test statistic $z$ (standardized $\hat p$).
4. Find p-value from the standard normal distribution.
5. Decision: compare p-value to significance level $\alpha$ (e.g., 0.05).
6. Contextual conclusion: interpret in plain language.

Standardization and the z Test Statistic

• Formula $z = \dfrac{\hat p - p<em>0}{\sqrt{\dfrac{p</em>0(1-p_0)}{n}}}$
  • Numerator: observed difference between sample and hypothesized proportions.
  • Denominator: SE computed with $p<em>0$ (because under $H</em>0$, $p=p_0$).
• Interpretation
  • $z = 2$ ⇒ observed $\hat p$ lies 2 SEs above $p_0$.
  • Links directly to the normal curve & 68–95–99.7 rule.

Visual Recall of 68–95–99.7 Rule

• Within $\pm1$ SE: ~68 % of statistics.
• Within $\pm2$ SE: ~95 %.
• Within $\pm3$ SE: ~99.7 %.
• Helps build intuition; exact p-values often require software.

Using R’s pnorm() for Normal Probabilities

• Syntax: pnorm(q, mean = 0, sd = 1, lower.tail = TRUE)
  • q = quantile (z-score)
  • lower.tail = TRUE (default) ⇒ $P(Z \le q)$.
  • lower.tail = FALSE ⇒ P(Z > q) (right tail).
• Tail selection depends on the alternative hypothesis:
  • Left-tailed (<): p-value = pnorm(z, ... , lower.tail = TRUE).
  • Right-tailed (>): p-value = pnorm(z, ... , lower.tail = FALSE).
  • Two-tailed ($\neq$): p-value = 2 * pnorm(|z|, lower.tail = FALSE) (or multiply the smaller tail by 2).

Worked Examples

1 Dolphin Communication (Buzz & Doris)

• Hypotheses: H0!: p = 0.50 \;\text{vs}\; HA!: p > 0.50.
• Data: $n = 16,\; \hat p = 15/16 = 0.9375$.
• Test statistic
  $z = \dfrac{0.9375-0.50}{\sqrt{0.50(1-0.50)/16}} = 3.50$.
• p-value (right tail)

  pnorm(3.50, mean = 0, sd = 1, lower.tail = FALSE)  # ≈ 0.00023

⇒ $p \approx 0.00023$ (highly significant).

2 Community Recycling Rate

• Hypotheses: $H<em>0!: p = 0.70 \;\text{vs}\; H</em>A!: p \neq 0.70$.
• Data: $n = 800,\; \hat p = 530/800 = 0.6625$.
• Test statistic
  $z = \dfrac{0.6625-0.70}{\sqrt{0.70(1-0.70)/800}} \approx -2.315$.
• p-value (two tails)

  2 * pnorm(-2.315, mean = 0, sd = 1, lower.tail = TRUE)  # ≈ 0.0206

⇒ $p \approx 0.0206$ (significant at $\alpha=0.05$ but not at stricter levels like 0.01).

Interpreting p-Values & Decisions

• Definition: p-value = probability, under $H_0$, of observing a statistic as extreme or more extreme than the one obtained.
• Direction matters:
  • $H<em>A: p < p</em>0$ ⇒ left tail only.
  • $H<em>A: p > p</em>0$ ⇒ right tail only.
  • $H<em>A: p \neq p</em>0$ ⇒ both tails (double the smaller tail).
• Decision rule at level $\alpha$
  • If $p \le \alpha$ ⇒ Reject $H_0$ ⇒ result is statistically significant.
  • If p > \alpha ⇒ Fail to reject $H_0$ ⇒ not statistically significant.

Practical & Conceptual Takeaways

• Once conditions hold, the normal model offers a fast, exact alternative to resampling.
• Checking assumptions (randomness & S–F) is not optional; violations may invalidate results.
• The SE built from $p_0$ reflects the variability expected under the null—a key distinction from confidence intervals.
• Software (e.g., R, pnorm) is usually needed for precise p-values beyond simple z scores of 1, 2, 3.
• Interpret findings in the context of the original research question, not just in terms of abstract numbers.