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

Course Logistics & Upcoming Deadlines

• Welcome to PSTAT 5LS
  • Current topic: Theory-Based Inference for p (beginning Slide 31)
  • Next topic: Inference for One Mean
• Deadlines
  • HW 3: Monday July 14 @ 11:59 PM
  • HW 4: Friday July 18 @ 11:59 PM
  • Exam 1: Wednesday July 16 (during lecture)
• Office Hours
  • Second OH this week: Monday @ 12 PM on Zoom

Exam 1 Information & Tips

• Coverage
  • Slide Sets 1 – 5
  • HW 1 – 3
• Format
  • $15\text{–}18$ multiple-choice + $2\text{–}4$ free-response
  • Write directly on the exam; formula sheet provided
• What to bring
  • Pencil/pen, calculator, photo ID (UCSB or other)
• Tips for success
  • Read each question carefully
  • Include context and units in explanations
  • Show complete work (except on MC questions)
  • Double-check that every part is answered

Transition from Hypothesis Testing to Estimation

• Hypothesis testing: evaluates evidence against claims about population parameters
• Estimation: uses sample statistics to approximate population parameters
  • Point estimate = single number → best guess of the parameter
  • For one proportion: $\hat p$ estimates $p$
  • Natural sampling variability → need a margin of error (MOE) to create a range of plausible values

Confidence Intervals – Core Idea

• Provide a range of plausible parameter values
• Generic structure
  • $\text{Point Estimate} \; \pm \; \text{Margin of Error}$
• Margin of Error
  • $\text{MOE} = (\text{multiplier}) \times (\text{standard error})$
  • Multiplier = critical value from a probability distribution (usually normal)
  • Standard error = estimated SD of the sampling distribution

Standard Error vs. Margin of Error (One Proportion)

• Standard Error (SE)
  • Measures typical variability in $\hat p$
  • $SE = \sqrt{\dfrac{p(1-p)}{n}}$ (unknown $p$ will later be replaced by $\hat p$)
• Margin of Error (MOE)
  • Adjusts SE for the desired confidence
  • $MOE = z^* \times SE$

Final CI Formula for One Proportion

• Because $p$ is unknown, plug in $\hat p$ for both point estimate and SE
• Confidence interval:
  $\hat p \; \pm \; z^* \times \sqrt{\dfrac{\hat p(1-\hat p)}{n}}$
• Higher confidence ⇒ larger $z^*$ ⇒ wider interval

Visualizing Multiple CIs

• Simulations show:
  • Each sample (black dot) has its own $\hat p$
  • Blue lines = corresponding CIs; red line = true $p$
  • Approximately the chosen % of intervals contain the red line
• Changing confidence level
  • 90 % → narrow
  • 95 % → medium
  • 99 % → wide

Choosing the Critical Value $z^*$

• Standard normal cut-offs (two-sided)
  • 90 % ⇒ $z^*=1.645$
  • 95 % ⇒ $z^*=1.960$
  • 98 % ⇒ $z^*=2.326$
  • 99 % ⇒ $z^*=2.576$
• Larger $z^*$ gives a bigger MOE to ensure higher confidence

What Does “Confidence” Mean?

• The confidence level applies to the method, not to a particular interval
• Example: 95 % confidence ⇒ if we repeated sampling infinitely, ~95 % of computed intervals would include $p$
• Each specific CI either contains $p$ or it doesn’t (probability 0 or 1 after data are collected)

Conditions for Constructing a CI for $p$

1. Independence
   • Random sampling/assignment OR sampling fraction <10\% of population
2. Success–Failure (S–F) Condition
   • Need at least 10 expected successes and 10 expected failures
   • For CIs use $\hat p$: $n\hat p \ge 10$ and $n(1-\hat p) \ge 10$

Three-Step CI Procedure

1. Check conditions
2. Calculate CI with correct $z^*$ and SE
3. Interpret in context (mention population, parameter, and confidence level)

Worked Example 1 – Gallup Math Feelings

• Survey: $n = 5136$ U.S. adults; $\hat p = 0.47$ reported only positive feelings about math
• Conditions
  • Independence: random sample ⇒ satisfied
  • S–F: $n\hat p = 5136(0.47) = 2413.92$ ≥ 10, and $n(1-\hat p)=2722.08$ ≥ 10 ⇒ satisfied
• 95 % CI
  • $SE = \sqrt{\dfrac{0.47(0.53)}{5136}} = 0.00696425$
  • $MOE = 1.960 \times 0.00696425 = 0.0136$
  • Interval: $0.47 \pm 0.0136 \; \Rightarrow \; (0.4564,\;0.4836)$
• Interpretation (proper wording)
  • “We are 95 % confident that between 45.64 % and 48.36 % of all U.S. adults have only positive feelings about math.”
  • Cannot state probability that $p$ is in this specific interval — it either is or isn’t.

Worked Example 2 – Gen Z & Water Pollution

• Parameter: $p=$ proportion of all Gen Zers (ages 12–27) who say protecting waters from pollution is very important
• Data: $x=2096,\;n=2832,\;\hat p = 0.740113$
• Conditions
  • Independence: random sample ⇒ satisfied
  • S–F: $n\hat p = 2832(0.740113)=2096\,(\ge10)$ and $n(1-\hat p)=736\,(\ge10)$ ⇒ satisfied
• 98 % CI (requires $z^*=2.326$)
  • $SE = \sqrt{\dfrac{0.740113(1-0.740113)}{2832}} = 0.00918$ (approx.)
  • $MOE = 2.326 \times 0.00918 \approx 0.0214$
  • CI: $0.7401 \pm 0.0214 \Rightarrow (0.7209,\;0.7593)$
• Decision Questions
  • 95 % CI would be narrower (smaller $z^*$)
  • Is more than 75 % supported? CI upper bound 0.7593, lower 0.7209 → interval includes 0.75, so cannot conclusively claim > 75 % at 98 % confidence.
• Multiple-choice interpretation (Slide 47)
  • Correct statements: b, c, e

Using R – prop_test()

prop_test(x = 2096, n = 2832, conf.level = 0.98)
## 1-sample proportions test without continuity correction
## Z = 25.556, p-value < 2.2e-16
## 98 percent confidence interval:
##  0.7209409 0.7592851
## sample estimates:
##  p 
## 0.740113

• Function automatically calculates CI, $Z$ statistic, and $p$-value (default null $p_0=0.5$)
• Ensures reproducibility and quick checks during analysis

Conceptual Take-Aways

• MOE grows with both $z^*$ and $SE$
• CIs quantify uncertainty; they do not guarantee the parameter lies inside
• Always verify assumptions before trusting a CI

Looking Forward

• Having formalized theory-based CIs & tests for one proportion, the course will extend these concepts to means (Slide 51 & future lectures).