Sampling Methods

📊 Collecting Data & Study Design

Part 1 of 7 — Observational Studies vs. Experiments

Two Ways to Gather Data

🔑 Key Principle: Only a well-designed experiment can establish a cause-and-effect relationship.

Observational Studies

In an observational study, researchers simply observe and record data without manipulating any variables.

Types:

• Retrospective — looks at past data (e.g., medical records)
• Prospective — follows subjects forward in time (e.g., tracking diet over 10 years)

Example: Studying whether coffee drinkers have lower rates of depression by surveying existing habits.

⚠️ Confounding variables lurk in observational studies. Maybe coffee drinkers also exercise more — that could be the real reason for lower depression.

Experiments

In an experiment, the researcher imposes treatments on subjects and measures the response.

Key Elements:

Concept Check 🎯

Identifying Study Components 🧮

A pharmaceutical company randomly assigns 200 patients to receive either a new drug or a placebo, then measures blood pressure after 8 weeks.

1) What is the explanatory variable? (drug/placebo or blood pressure)

2) How many treatment groups are there?

3) Can this study establish causation? (yes or no)

Study Design Classification 🔽

🎲 Sampling Methods

Part 2 of 7 — How to Select a Representative Sample

Why Sampling Matters

We rarely have the resources to study an entire population. Instead, we take a sample and use it to make inferences about the population.

🔑 Goal: The sample should be representative of the population — every individual should have a known chance of being selected.

Probability Sampling Methods

⚠️ Sources of Bias

Part 3 of 7 — What Can Go Wrong

Types of Bias

🔑 A biased sampling method will produce biased results no matter how large the sample.

Reducing Bias

• Use random selection to avoid selection bias
• Follow up with to reduce nonresponse bias

🔬 Principles of Experimental Design

Part 4 of 7 — Control, Randomize, Replicate, Block

Four Principles of Good Experiments

🔑 Random assignment → reduces confounding → supports causal claims

Completely Randomized Design

The simplest experimental design:

1. Pool all experimental units
2. Randomly assign each to a treatment group
3. Compare responses

Example: 60 patients randomly assigned to Drug A (30) vs. Placebo (30)

Randomized Block Design

🎰 Random Variables & Expected Value

Part 5 of 7 — Discrete Random Variables

What Is a Random Variable?

A random variable $X$ assigns a numerical value to each outcome of a random process.

Probability Distribution of a Discrete RV

A table showing all values and their probabilities:

🛠️ Problem-Solving Workshop

Part 6 of 7 — Applying Study Design Concepts

Strategy for AP Statistics Study Design Questions

1. Identify the study type — Is a treatment being imposed? If yes → experiment. If no → observational.
2. Check for bias — Look for selection bias, nonresponse, response bias, voluntary response.
3. Identify confounding — What other variables could explain the observed relationship?
4. Evaluate design — Does it use random assignment? Control group? Blinding? Blocking?

Worked Example 1

Scenario: A school wants to test whether a new math curriculum improves test scores. They implement the new curriculum in School A and keep the old one in School B, then compare end-of-year scores.

Analysis:

• ❌ Not a randomized experiment — schools were not randomly assigned
• ⚠️ Confounding: Schools may differ in student demographics, teacher quality, funding
• 🔧 Better design: Randomly assign classrooms within the SAME school to old vs. new curriculum

Worked Example 2

Scenario: Researchers want to know if a new drug lowers cholesterol. They recruit 200 volunteers, randomly assign 100 to the drug and 100 to a placebo, and measure cholesterol after 3 months. Neither patients nor doctors know who gets which pill.

Analysis:

• ✅ Randomized experiment — can establish causation
• ✅ Control group (placebo) — accounts for placebo effect
• ✅ Double-blind — reduces bias from expectations
• ✅ Replication — 100 per group is adequate

Workshop Problems 🎯

📋 Review & Applications

Part 7 of 7 — Comprehensive Review

Key Concepts Summary