AP Statistics (AP Stats): Blocking and Matched Pairs (and why they reduce variability)

AP Statistics – Blocking and Matched Pairs (and why they reduce variability)

AP Statistics: Blocking and Matched Pairs – Exam-Ready Study Guide

What This Is

Blocking and matched pairs are experimental design techniques that reduce variability by controlling for confounding variables. Blocking groups similar experimental units together (e.g., testing a new fertilizer on plots with similar soil types), while matched pairs compare two treatments on the same subject or very similar subjects (e.g., measuring blood pressure before and after a drug). These methods are essential on the AP exam because they improve the precision of experiments, leading to more reliable conclusions. Expect FRQs where you must identify when to block, analyze matched-pairs data, or explain why blocking reduces variability.

Key Terms & Formulas

• Blocking: Grouping experimental units into blocks (homogeneous groups) to control for variability from confounding variables. Example: Testing a new teaching method by blocking students by grade level.
• Matched Pairs Design: A special case of blocking where each subject receives both treatments (e.g., before/after) or pairs of similar subjects are compared. Example: Testing two sunscreens on the same person’s left and right arms.
• Paired t-test (Matched Pairs t-test):
• H?: ?_d = 0 (no difference in means)
• H?: ?_d-0 (or ?_d > 0 or ?_d < 0)
• Test statistic: t = (x?_d – 0) / (s_d / ?n), where x?_d = mean of differences, s_d = standard deviation of differences, n = number of pairs.
• Calculator command: T-Test (L1 = differences, = 0, tails = 2/1).
• Degrees of Freedom (df) for Paired t-test: df = n – 1 (where n = number of pairs).
• Two-Sample t-test vs. Paired t-test: Use paired t-test when data is matched; use two-sample t-test when groups are independent.
• Randomized Block Design: Randomly assign treatments within each block to reduce variability. Example: Testing three diets on blocks of mice with similar weights.
• Confounding Variable: A variable that influences both the explanatory and response variables, making it hard to isolate the effect of the treatment. Example: Testing a new study method but not accounting for prior student ability.
• Reducing Variability: Blocking/matched pairs reduce the standard error by controlling for extraneous variability, making it easier to detect treatment effects.
• Calculator Tip for Paired Data: Enter differences into L1, then run T-Test (not 2-SampTTest).
• Interpretation of p-value (Paired t-test): If p-value < ?, reject H? and conclude there is a significant difference between treatments.

Step-by-Step / Process Flow

How to Solve a Matched Pairs FRQ:
1. Identify the Design: - Is the data paired (same subjects before/after, or matched pairs)? If yes, use a paired t-test. - If not, check if blocking was used (e.g., "subjects were blocked by age").

1. State Hypotheses:
2. H?: ?_d = 0 (no difference in means)

3. H?: ?_d-0 (or > 0 or < 0, depending on context).

4. Check Conditions:

5. Random: Data comes from a random sample or randomized experiment.
6. Normal/Large Sample: Either the differences are roughly normal (check with a histogram/boxplot) or n-30 (CLT).

7. Independent: Differences are independent (usually satisfied if subjects are randomly selected).

8. Compute the Test Statistic:

9. Calculate differences (Treatment A – Treatment B or Before – After).
10. Enter differences into L1 on TI-84.

11. Run T-Test (Stats-TESTS-2:T-Test, select "Data," = 0, List = L1).

12. Find the p-value:

13. The calculator gives the p-value. Compare to? (usually 0.05).

14. Make a Conclusion in Context:

15. If p-value < ?: "Reject H?. There is convincing evidence that [treatment] has an effect."
16. If p-value-?: "Fail to reject H?. There is not convincing evidence that [treatment] has an effect."

Example FRQ Setup: "A researcher wants to test if a new energy drink improves reaction time. 20 volunteers take a reaction time test before and after drinking the energy drink. The differences (After – Before) are recorded. Should the researcher use a paired t-test or a two-sample t-test? Justify your answer."

Answer: Paired t-test, because the same subjects are measured before and after (matched pairs).

Common Mistakes

• Mistake: Using a two-sample t-test for matched pairs data.

• Correction: Use a paired t-test when data is matched (same subjects or pairs). Two-sample t-tests assume independent groups, which is not true for matched pairs.

• Mistake: Forgetting to calculate differences before running the test.

• Correction: Always compute differences (e.g., After – Before) and enter them into L1 for a paired t-test.

• Mistake: Misidentifying the alternative hypothesis (e.g., using ?_d > 0 when the context suggests ?_d < 0).

• Correction: Read the problem carefully! If testing if a drug lowers blood pressure, H?: ?_d < 0 (where d = Before – After).

• Mistake: Not checking normality for small samples (n < 30).

• Correction: Always check if the differences are roughly normal (histogram/boxplot) or state that n-30 (CLT applies).

• Mistake: Confusing blocking with stratified sampling.

• Correction: Blocking is used in experiments to control variability; stratified sampling is used in surveys to ensure representation.

AP Exam Insights

• FRQs often ask you to:
• Explain why blocking reduces variability (e.g., "Blocking by age reduces variability from age differences, making it easier to detect the effect of the treatment.").
• Compare matched pairs vs. independent groups (e.g., "Why is a matched pairs design better than a two-sample design for this scenario?").

• Interpret p-values in context (e.g., "A p-value of 0.03 means there is a 3% chance of observing a mean difference this extreme if the energy drink had no effect.").

• Tricky Distinctions:

• Paired t-test vs. Two-Sample t-test: Paired = same subjects/pairs; two-sample = independent groups.
• Blocking vs. Stratifying: Blocking is for experiments; stratifying is for sampling.

• Reducing Variability vs. Eliminating Bias: Blocking reduces variability but does not eliminate bias (randomization does).

• Calculator Pitfalls:

• Using 2-SampTTest instead of T-Test for paired data.
• Forgetting to enter differences into L1 before running the test.
• Misinterpreting the p-value (e.g., thinking a p-value of 0.01 means 1% of the data supports H?).

Quick Check Questions

1. Multiple Choice: A researcher tests a new fertilizer by applying it to 10 plots of corn and comparing the yield to 10 untreated plots. The plots are not matched in any way. Which test should be used?
2. (A) Paired t-test
3. (B) Two-sample t-test
4. (C) Chi-square test

5. (D) One-sample z-test Answer: (B) Two-sample t-test (groups are independent).

6. FRQ Part: A study measures the effect of a new teaching method on test scores. 30 students take a pre-test and post-test. The differences (Post – Pre) are recorded.

7. (a) Should the researcher use a paired t-test or a two-sample t-test? Explain.
8. (b) State the hypotheses for the test. Answer:
9. (a) Paired t-test, because the same students are measured before and after (matched pairs).

10. (b) H?: ?_d = 0 (no difference in means), H?: ?_d > 0 (post-test scores are higher).

11. Multiple Choice: Why is blocking used in experiments?

12. (A) To increase the sample size
13. (B) To reduce variability from confounding variables
14. (C) To ensure random assignment
15. (D) To eliminate all bias Answer: (B) To reduce variability from confounding variables.

Last-Minute Cram Sheet

1. Blocking = Group similar experimental units to reduce variability.
2. Matched Pairs = Same subjects or pairs (use paired t-test).
3. Paired t-test hypotheses: H?: ?_d = 0, H?: ?_d-0 (or >/< 0).
4. Paired t-test df = n – 1 (n = number of pairs).
5. Calculator for paired t-test: Enter differences into L1, then T-Test ( = 0).
6. Check conditions: Random, Normal/Large Sample (n-30 or check histogram), Independent.
7. Blocking-Stratifying (blocking = experiments, stratifying = surveys).
8. Two-sample t-test = Independent groups; paired t-test = matched data.
9. Always calculate differences for paired data!
10. Don’t use 2-SampTTest for paired data!