AP Statistics (AP Stats): Completely Randomized Design and Randomized Block Design

AP Statistics – Completely Randomized Design and Randomized Block Design

AP Statistics Study Guide: Completely Randomized Design & Randomized Block Design

What This Is

This topic covers how to design experiments to compare treatments while controlling for variability. A completely randomized design (CRD) assigns subjects randomly to treatments, while a randomized block design (RBD) groups similar subjects (blocks) first, then randomizes within blocks to reduce variability. These designs are essential for establishing causation (e.g., testing if a new fertilizer increases crop yield, comparing the effectiveness of two teaching methods). The AP exam tests your ability to identify designs, justify their use, and analyze results.

Key Terms & Formulas

• Completely Randomized Design (CRD): Subjects are randomly assigned to treatments with no grouping. Used when subjects are homogeneous.
• Randomized Block Design (RBD): Subjects are grouped into blocks based on a variable that may affect the response, then randomly assigned to treatments within blocks. Reduces variability.
• Blocking variable: A variable (e.g., age, gender) used to group subjects in RBD to control for its effect.
• Experimental units: The smallest collection of individuals to which treatments are applied (e.g., patients, plots of land).
• Response variable: The outcome measured (e.g., blood pressure, crop yield).
• Explanatory variable (factor): The variable manipulated (e.g., drug vs. placebo, fertilizer type).
• Treatment: A specific condition applied to experimental units (e.g., "Drug A," "No fertilizer").
• Control group: A baseline group that receives no treatment or a standard treatment for comparison.
• Replication: Using enough experimental units to distinguish treatment effects from random variation.
• Random assignment: Ensures each unit has an equal chance of receiving any treatment, balancing lurking variables.
• Matched pairs design: A special case of RBD where blocks consist of pairs of similar units (e.g., twins, left/right arms).
• ANOVA (Analysis of Variance): A statistical test used to compare means across multiple groups (not required for AP Stats, but know it’s used for CRD/RBD).

Step-by-Step / Process Flow

1. Identify the Design

• CRD: If subjects are randomly assigned to treatments with no grouping, it’s a CRD.
• RBD: If subjects are first grouped by a blocking variable (e.g., age, gender), then randomly assigned within blocks, it’s an RBD.
• Matched pairs: If blocks are pairs of similar units (e.g., before/after measurements on the same person), it’s a matched pairs design.

2. Justify the Design Choice

• CRD: Use when subjects are similar or blocking isn’t necessary (e.g., testing a new drug on a homogeneous population).
• RBD: Use when a variable (e.g., gender, soil type) might affect the response. Blocking reduces variability and makes treatment effects clearer.
• Example justification: "We used a randomized block design with gender as the blocking variable because males and females may respond differently to the drug, and blocking reduces variability in the response."

3. Describe Random Assignment

• CRD: Use a random number generator or table to assign subjects to treatments (e.g., randInt(1, 3, 10) on TI-84 to assign 10 subjects to 3 treatments).
• RBD: First, group subjects into blocks. Then, within each block, randomly assign treatments (e.g., for 2 treatments and 3 blocks, use randInt(1, 2, n) for each block).
• Matched pairs: Randomly assign one treatment to each unit in the pair (e.g., flip a coin to decide which twin gets the drug).

4. Analyze Results (FRQ Tips)

• Compare means/medians: Use side-by-side boxplots or summary statistics (e.g., 1-Var Stats on TI-84).
• State hypotheses (if testing):
• H?: No difference in means (e.g., = = ).
• H?: At least one mean is different.
• Check conditions:
• Random assignment (to generalize to the population if sampling was random).
• Normality: Check if sample sizes are large (n-30) or if data is roughly symmetric (use graphs).
• Equal variance: Spread of data should be similar across groups (check boxplots).
• Compute test statistic (if required): For AP Stats, you’ll usually interpret output (e.g., p-value from a computer output).
• Make a conclusion in context:
• If p-value <? (e.g., 0.05), reject H?: "There is convincing evidence that the treatments differ."
• If p-value-?, fail to reject H?: "There is not convincing evidence that the treatments differ."

5. Interpret Blocking (RBD Only)

• Explain how blocking reduces variability: "Blocking by gender controlled for differences in response between males and females, making the treatment effect clearer."
• Compare to CRD: "A CRD might have shown no difference due to high variability, but blocking revealed a significant effect."

Common Mistakes

Mistake 1: Confusing random sampling and random assignment.

• Correction: Random sampling (selecting subjects from a population) is for generalization. Random assignment (assigning subjects to treatments) is for causation. Experiments use random assignment; observational studies use random sampling.
• Why? The AP exam loves to test this distinction (e.g., "Can we generalize to all high school students?" vs. "Can we conclude the drug caused the effect?").

Mistake 2: Forgetting to justify blocking.

• Correction: Always explain why you blocked (e.g., "We blocked by age because older patients may respond differently to the drug.").
• Why? FRQs often ask, "Why did the researchers use a randomized block design?" A vague answer like "to reduce variability" isn’t enough.

Mistake 3: Misidentifying the design.

• Correction: If subjects are grouped before random assignment, it’s RBD. If not, it’s CRD. Matched pairs is a type of RBD.
• Why? The AP exam may describe an experiment and ask, "Is this a completely randomized design or a randomized block design?"

Mistake 4: Ignoring the response variable.

• Correction: Always state the response variable (e.g., "The response variable is crop yield in pounds.").
• Why? FRQs often ask, "What is the response variable in this experiment?" Missing this costs points.

Mistake 5: Not describing random assignment clearly.

• Correction: For full credit, describe how random assignment was done (e.g., "Assign each subject a unique number, then use a random number generator to assign them to treatments.").
• Why? The AP rubric rewards specific details.

AP Exam Insights

1. FRQs Love Design Justification

• Expect a question like: "Why did the researchers use a randomized block design instead of a completely randomized design?" Your answer must mention the blocking variable and how it reduces variability.
• Example: "Blocking by soil type controlled for differences in fertility, making the effect of the fertilizer clearer."

2. Matched Pairs is a Favorite

• The AP exam often includes matched pairs (e.g., before/after measurements, twins, left/right hands). Know how to:
• Identify it (blocks of size 2).
• Analyze it (compute differences, then use a 1-sample t-test or CI).

3. Calculator Tips for Random Assignment

• TI-84 for CRD: randInt(1, k, n) assigns n subjects to k treatments.
• Example: randInt(1, 3, 20) assigns 20 subjects to 3 treatments.
• TI-84 for RBD: Do randInt separately for each block.
• Example: For 2 treatments and 3 blocks of 4 subjects each, run randInt(1, 2, 4) three times.

4. Tricky Distinctions

• CRD vs. RBD: CRD is simpler but may have more variability. RBD reduces variability but requires more planning.
• Blocking vs. Stratifying: Blocking is for experiments; stratifying is for sampling. Don’t mix them up!
• Causation: Only experiments with random assignment can establish causation. Observational studies cannot.

Quick Check Questions

1. Multiple Choice

A researcher wants to test whether a new fertilizer increases crop yield. She divides 60 plots of land into 3 groups based on soil type (clay, sand, loam) and then randomly assigns 20 plots within each group to receive either the new fertilizer or the old fertilizer. What type of design is this? - (A) Completely randomized design - (B) Randomized block design - (C) Matched pairs design - (D) Observational study

Answer: (B) Randomized block design. The plots are first grouped by soil type (blocking variable), then randomly assigned to treatments.

2. FRQ Part

A company tests two new website designs (A and B) to see which leads to more purchases. They randomly assign 100 visitors to each design. However, they suspect that age might affect the results, so they first group visitors into three age categories (18-25, 26-40, 41+), then randomly assign visitors within each group to designs A or B.

a) Identify the design and explain why it was used. b) What is the response variable in this experiment?

Answer: a) This is a randomized block design. It was used to control for the effect of age on purchases, reducing variability and making the comparison of designs clearer. b) The response variable is the number of purchases (or "purchase amount") made by each visitor.

3. Multiple Choice

Which of the following is a benefit of using a randomized block design instead of a completely randomized design? - (A) It allows for generalization to a larger population. - (B) It reduces variability in the response variable. - (C) It ensures that all subjects receive the same treatment. - (D) It eliminates the need for random assignment.

Answer: (B) It reduces variability in the response variable. Blocking controls for a variable that might affect the response, making treatment effects easier to detect.

Last-Minute Cram Sheet

1. CRD: Randomly assign subjects to treatments with no grouping.
2. RBD: Group subjects by a blocking variable, then randomly assign within blocks.
3. Matched pairs: Blocks of size 2 (e.g., twins, before/after).
4. Blocking variable: A variable (e.g., age, gender) used to group subjects in RBD.
5. Random assignment: Use randInt(1, k, n) on TI-84 for CRD; do separately for each block in RBD.
6. Response variable: The outcome measured (e.g., crop yield, test scores).
7. Justify blocking: "Blocking by [variable] reduces variability in the response."
8. Causation: Only experiments with random assignment can establish causation.
9. Don’t confuse blocking (experiments) with stratifying (sampling).
10. Always describe how random assignment was done for full FRQ credit.