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

Simple Random Sample (SRS)

An SRS of size $n$ means every set of $n$ individuals has an equal chance of being the chosen sample.

How to do it:

1. Assign each individual a number
2. Use a random number generator (or table) to select $n$ numbers
3. The corresponding individuals form your sample

Example: To sample 50 students from a school of 800, number them 001–800 and use a random number table to pick 50 numbers.

Stratified Random Sampling

1. Divide the population into strata (groups that are similar within)
2. Take an SRS from each stratum
3. Combine the SRS results

When to use: When the population has distinct subgroups (e.g., grade levels, gender, income brackets)

Example: A school with 400 freshmen and 300 seniors → sample 40 freshmen (SRS) and 30 seniors (SRS).

Sampling Methods Check 🎯

Sampling Calculations 🧮

A school has 600 students: 200 freshmen, 150 sophomores, 150 juniors, and 100 seniors. A stratified sample of 60 students is taken proportionally.

1) How many freshmen should be sampled? (200/600 × 60)

2) How many seniors should be sampled? (100/600 × 60)

3) In a systematic sample of every 10th student from a list of 600, how many students are in the sample?

Identify the Sampling Method 🔽

⚠️ 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 convenience samples and undercoverage
• Follow up with nonrespondents to reduce nonresponse bias
• Use neutral wording and anonymous surveys to reduce response bias
• Match sample demographics to population demographics

Bias Identification 🎯

Bias Analysis 🧮

1) A survey is conducted at a shopping mall on a Wednesday afternoon. Name the type of bias this introduces. (undercoverage, response, nonresponse, or voluntary response)

2) Only 15 of 100 mailed surveys are returned. This is an example of what type of bias?

3) True or false: Doubling the sample size from 500 to 1000 will eliminate bias. (true or false)

🔬 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

When you know a variable (like age or gender) might affect results:

1. Block subjects by that variable
2. Randomly assign treatments within each block

Example: Block by gender (male/female), then randomly assign drug/placebo within each gender block

Matched Pairs Design

A special case of blocking where each block has exactly 2 units (or the same person gets both treatments):

• Two matched subjects: one gets treatment, one gets control
• Same subject: each person serves as their own control (crossover design)

Design Principles 🎯

Identify the 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:

Requirements: All probabilities are between 0 and 1, and they sum to 1: $\sum P(X = x) = 1$

Expected Value (Mean)

$E(X) = \mu_X = \sum x_i \cdot P(X = x_i)$

Example: $E(X) = 0(0.1) + 1(0.3) + 2(0.4) + 3(0.2) = 0 + 0.3 + 0.8 + 0.6 = 1.7$

🔑 The expected value is the long-run average — if you repeated the process many times, the average outcome would approach $E(X)$.

Variance and Standard Deviation

$\text{Var}(X) = \sigma_X^2 = \sum (x_i - \mu_X)^2 \cdot P(X = x_i)$

$\sigma_X = \sqrt{\text{Var}(X)}$

Random Variables Check 🎯

Expected Value Calculations 🧮

A game costs $5 to play. You roll a die: if you get a 6, you win $20; otherwise you win nothing.

1) What is $P(\text{win})$? Express as a decimal (round to 2 places).

2) What is the expected payout (not profit)? Round to nearest cent.

3) What is the expected profit per game? (payout minus cost, round to nearest cent)

🛠️ 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 undercoverage, 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

Comprehensive Review 🎯

Final Review 🔽