Topic Four: Sampling Methods, Margin of Error, and Exam Logistics

Announcements and Exam Logistics

  • Assignments and announcements due by Sunday.
  • Extra credit opportunity: please participate; closes when Exam One closes (11:59 PM on next Tuesday).
  • Exam One details:
    • Opens on Sunday (times mentioned: noon and 12:30 are noted in the transcript).
    • You have two and a half days to complete after opening.
    • Duration: 75 minutes.
    • Open book and open notes.
    • You are expected to work on your own.
    • Requires a laptop browser and a working webcam during the exam.
  • Practice Exam One: also due at the same time as Exam One.
  • Participation assessment: mandatory and part of your grade. You can take it multiple times to improve.
  • Score for the participation assessment: up to 10 points.
  • Office hours today: class until noon.
  • This Friday: Exam One Q&A session (not a full content review of topics 1–4, but a survey of questions).
    • Submit questions or topics you want covered via the posted survey link under Topic Four → Exam One.
    • Please submit questions by tomorrow.
  • Any questions or concerns?
  • Friday’s plan: no full lecture; there will be an Exam One Q&A session.

This Week’s Schedule and Topics

  • Today’s lecture: cover everything for Topic Four.
  • Friday: Exam One Q&A session. Submit questions by tomorrow if you want specific topics covered.

Topic Four: Sampling Methods Overview

  • There are two broad families of sampling methods:
    • Probability sampling methods: pursued to sample a population with known probabilities.
    • Nonprobability sampling methods: to be avoided when possible to minimize bias, though may be used in some contexts.

Probability Sampling Methods (Five Methods)

  • Simple random sample:
    • Start with a population pool and randomly sample n individuals.
    • Visual representation: black dots indicate randomly sampled individuals from the population.
  • Systematic random sampling:
    • Requires a sampling frame (a population list).
    • Choose a starting point and then select every k-th individual (e.g., start at the 5th, sample every 6th).
    • Another example: start at the 1st individual and sample every 10th.
  • Stratified random sampling:
    • Divide the population into strata (subgroups) based on key variables (e.g., year in college: freshman, sophomore, junior, senior).
    • Randomly sample from each stratum.
  • Cluster sampling:
    • Population is divided into clusters; instead of sampling individuals, randomly sample entire clusters.
    • Then collect data from all individuals within selected clusters.
    • Example in transcript: a 5×5 box grid representing clusters; randomly select several clusters and sample everyone in those clusters.
  • Multistage sampling:
    • Uses two or more procedures of probability sampling across stages.
    • Example from transcript:
    • Stage 1: cluster sampling to select several clusters.
    • Stage 2: within each selected cluster, perform simple random sampling to pick individuals.
    • This can combine cluster + simple random sampling (or systematic sampling) across stages.

Nonprobability Sampling Methods

  • Convenience sampling:
    • Sample from whoever is accessible (e.g., friends, family, campus flyers, online postings).
    • Practical and common due to resource constraints; often used for ease and speed.
  • Voluntary response sampling:
    • Participants are those who volunteer to participate.

Biases with Nonprobability Sampling (Illustrative Points from Transcript)

  • Survivorship bias/examples discussed:
    • Quote 1: “Ninety percent of all deaths resulting from our own roles, the victim was the one who instigated the fights.”
    • Issue: people who died cannot respond to surveys; survivors may bias responses.
    • Quote 2: “In March 1991, the New York Times reported that eighty-eight percent of all women wear shoes at least one size too small,” based on podiatry data.
    • Issue: only podiatrists see feet problems, so sample is not representative; this is not a representative sampling frame.
  • These examples illustrate survivorship bias and sampling bias in nonprobability samples, and why representativeness matters for generalizing to the population.
  • Additional point: some topics cannot be studied directly if the population cannot be accessed (e.g., people who have died); researchers must work with accessible populations and acknowledge limitations.

Sampling Error and Margin of Error (MOE)

  • What is sampling error?
    • It is the error (or bias) that arises from the sampling procedure itself, causing the sample statistic to differ from the true population parameter.
    • Types mentioned in transcript:
    • Random sampling error: the portion of sampling error that occurs even when using a random sampling process (unavoidable to some extent due to chance).
    • Regular sampling error: (not explicitly defined in detail in transcript; included as part of the three types listed).
    • Non-sampling error: errors not due to the sampling procedure (e.g., nonresponse, missing data, measurement error).
  • Importance of margin of error (MOE):
    • MOE is a statistic that helps account for random sampling error and provides a range within which the population parameter is expected to lie most of the time (e.g., 95% of the time).
    • If an estimate is p-hat, the MOE defines the confidence interval around p-hat.
    • A larger sample size generally reduces MOE (n increases → MOE decreases).
  • MOE formulas (as discussed in transcript and standard practice):
    • Simple approximate MOE (proportion form, for a rough rule of thumb):
      extMOE≈frac1n(as a proportion)ext{MOE} \approx \\frac{1}{\sqrt{n}} \quad \text{(as a proportion)}
    • To express as a percentage:
      MOE%≈frac100n\text{MOE}_{\%} \approx \\frac{100}{\sqrt{n}} \\%.
    • More precise MOE using a z-score and p-hat:
      MOE=zα/2 p^(1−p^)n.\text{MOE} = z_{\alpha/2} \, \sqrt{\frac{\hat{p} (1-\hat{p})}{n}}.
    • For a 95% confidence level, typically z0.025≈1.96z_{0.025} \approx 1.96.
  • Interpretation of MOE and confidence intervals:
    • The confidence interval (CI) for a proportion is:
      p^±MOE.\hat{p} \pm \text{MOE}.
    • Saying “95% confidence” means that if we repeated the sampling process many times, approximately 95% of the computed intervals would contain the true population parameter.
    • Larger samples yield smaller MOE and narrower confidence intervals.
  • Worked examples from transcript:
    • Example 1: n = 400, observed proportion  p^=0.73\,\hat{p} = 0.73 (73%).
    • MOE (simple rule of thumb) = 100400=5%.\frac{100}{\sqrt{400}} = 5\%.
    • 95% CI (approximate) = 0.73±0.05⇒[0.68,0.78].0.73 \pm 0.05 \Rightarrow [0.68, 0.78].
    • Example 2: n = 750, Clinton = 46%, Trump = 43%.
    • MOE ≈ 100750≈3.7%.\frac{100}{\sqrt{750}} \approx 3.7\%.
    • Clinton 95% CI: [0.46−0.037, 0.46+0.037]=[0.423,0.497]⇒[42.3%,49.7%].[0.46 - 0.037, \, 0.46 + 0.037] = [0.423, 0.497] \Rightarrow [42.3\%, 49.7\%].
    • Trump 95% CI: [0.43−0.037, 0.43+0.037]=[0.393,0.467]⇒[39.3%,46.7%].[0.43 - 0.037, \, 0.43 + 0.037] = [0.393, 0.467] \Rightarrow [39.3\%, 46.7\%].
    • Overlap: The two confidence intervals overlap, so the data do not provide conclusive evidence that one candidate is ahead of the other at the population level.
    • Example 3: Increase to n = 7,500.
    • MOE ≈ 1007500≈1.15%(about 1.2%).\frac{100}{\sqrt{7500}} \approx 1.15\% \text{(about 1.2\%)}.
    • With the same point estimates (Clinton 46%, Trump 43%), the CIs would be approximately:
      • Clinton: [0.46−0.0115, 0.46+0.0115]=[0.4485,0.4715]⇒[44.85%,47.15%].[0.46 - 0.0115, \, 0.46 + 0.0115] = [0.4485, 0.4715] \Rightarrow [44.85\%, 47.15\%].
      • Trump: [0.43−0.0115, 0.43+0.0115]=[0.4185,0.4415]⇒[41.85%,44.15%].[0.43 - 0.0115, \, 0.43 + 0.0115] = [0.4185, 0.4415] \Rightarrow [41.85\%, 44.15\%].
    • These intervals do not overlap (approximately), giving greater confidence about which candidate leads at the population level (though this conclusion depends on the exact data and assumptions).

Practical Takeaways

  • Margin of error quantifies the uncertainty due to sampling; bigger samples reduce MOE and tighten confidence intervals.
  • Overlapping confidence intervals suggest that the difference between estimates may not be statistically significant at the chosen confidence level.
  • When planning samples, consider both desired precision (MOE) and feasible sample sizes; the trade-off is larger samples require more resources but yield more precise estimates.
  • Remember that MOE and CI assume random sampling; nonprobability samples can yield biased estimates and invalidate the usual MOE/CI interpretations.

Exam Q&A Session (Logistics Reminder)

  • No lecture this Friday; instead, an Exam One Q&A session for students who did not attend.
  • Submit questions or topics ahead of time via the Topic Four → Exam One Q&A survey link.
  • Ensure you complete the Exam One and Practice Exam One by their due dates to avoid penalties.