AP Statistics (AP Stats): Paired t?test (Matched Pairs) – Analyze Differences

AP Statistics – Paired t?test (Matched Pairs) – Analyze Differences

AP Statistics: Paired t-test (Matched Pairs) – Study Guide

What This Is

A paired t-test (or matched-pairs t-test) is used to compare two related measurements—such as before-and-after results for the same subjects—to determine if there is a statistically significant difference. This is essential on the AP exam because it tests your ability to analyze dependent samples (unlike two-sample t-tests, which compare independent groups). Real-world example: A researcher wants to know if a new study technique improves students' test scores by comparing their scores before and after using the technique.

Key Terms & Formulas

• Paired Data: Two measurements taken on the same subjects (e.g., pre-test and post-test scores, left vs. right hand strength).
• Differences (d): For each pair, compute the difference (e.g., post-score – pre-score). The paired t-test analyzes these differences.
• Hypotheses:
• H?: = (or ?_d = 0)-No difference between the two conditions.
• H?: - (or ?_d-0)-There is a difference (two-tailed).
• H?: > (or ?_d > 0)-First condition is greater (one-tailed).
• H?: < (or ?_d < 0)-First condition is smaller (one-tailed).
• Test Statistic (t): [ t = \frac{\bar{d} - 0}{s_d / \sqrt{n}} ]
• d? = mean of the differences
• s_d = standard deviation of the differences
• n = number of pairs
• Degrees of Freedom (df): df = n – 1 (where n = number of pairs).
• Conditions for Inference (LINER for differences):
• Linear (data is paired, differences are meaningful).
• Independent: The pairs themselves must be independent (e.g., different subjects).
• Normal: The differences should be approximately normal (check with a histogram or NPP; n-30 is usually safe).
• Experimental design: Data must be paired (not independent samples).
• Random: Data should come from a random sample or randomized experiment.
• Confidence Interval for ?_d: [ \bar{d} \pm t^* \left( \frac{s_d}{\sqrt{n}} \right) ]
• Use invT(area to left, df) to find t* (e.g., for 95% CI, invT(0.975, df)).
• Calculator Commands (TI-84):
• 1-Var Stats (for differences): STAT-CALC-1-Var Stats (enter differences in L1).
• T-Test (for paired data): STAT-TESTS-T-Test (choose "Data" if differences are in a list, or "Stats" if you have d?, s_d, and n).
• T-Interval: STAT-TESTS-TInterval (for confidence intervals).

Step-by-Step / Process Flow

Follow these steps for a paired t-test FRQ:

1. State Hypotheses:
2. Define ?_d (mean difference) in context.

3. Write H?: ?_d = 0 and H?: ?_d-0 (or >/< 0 for one-tailed tests).

4. Check Conditions:

5. Paired Data: Confirm the data is matched (e.g., same subjects, before/after).
6. Independence: Pairs are independent (e.g., different students).
7. Normality: Check if differences are roughly normal (histogram/NPP) or n-30.

8. Random: Data comes from a random sample/experiment.

9. Compute Test Statistic:

10. Calculate differences for each pair.
11. Find d? (mean of differences) and s_d (standard deviation of differences).
12. Plug into the t-formula: [ t = \frac{\bar{d} - 0}{s_d / \sqrt{n}} ]

13. OR use TI-84: T-Test (enter differences in L1).

14. Find p-value:

15. Use tcdf(lower, upper, df) on TI-84 (e.g., for two-tailed, 2 * tcdf(abs(t), 1E99, df)).

16. OR use T-Test output.

17. Make a Conclusion in Context:

18. Compare p-value to ? (usually 0.05).
19. If p-?, reject H?-"There is convincing evidence that [alternative hypothesis in context]."

20. If p > ?, fail to reject H?-"There is not convincing evidence that [alternative hypothesis in context]."

21. (Optional) Confidence Interval:

22. If asked, compute a CI for ?_d using: [ \bar{d} \pm t^* \left( \frac{s_d}{\sqrt{n}} \right) ]
23. Interpret: "We are [95%] confident that the true mean difference in [context] is between [lower] and [upper]."

Common Mistakes

• Mistake: Treating paired data as independent (e.g., using a two-sample t-test instead of paired).

• Correction: Always check if data is paired (same subjects, before/after). If so, use a paired t-test.

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

• Correction: The paired t-test analyzes differences, not raw data. Subtract one measurement from the other for each pair.

• Mistake: Misstating hypotheses (e.g., writing H?: = instead of H?: ?_d = 0).

• Correction: Hypotheses should be about the mean difference (?_d), not two separate means.

• Mistake: Ignoring normality for small samples.

• Correction: If n < 30, check normality of differences (histogram/NPP). If skewed, the test may not be valid.

• Mistake: Using the wrong df (e.g., df = n? + n? – 2 for paired data).

• Correction: For paired t-tests, df = n – 1 (where n = number of pairs).

AP Exam Insights

• Tricky Distinction: Paired t-test vs. two-sample t-test.
• Paired: Same subjects, before/after (e.g., weight loss program).
• Two-sample: Independent groups (e.g., comparing two different classes).

• AP loves to test this! Always ask: "Are the data points related?"

• Common FRQ Setup:

• A scenario with before-and-after measurements (e.g., drug trials, training programs).
• Often includes a table of paired data (you must compute differences).

• May ask for hypotheses, conditions, test statistic, p-value, and conclusion.

• Calculator Pitfalls:

• Forgetting to enter differences into L1 before running T-Test.
• Using 2-SampTTest instead of T-Test (paired data is not two independent samples).

• Misinterpreting p-value output (e.g., one-tailed vs. two-tailed).

• Interpretation Traps:

• Confidence Interval: If 0 is not in the CI, reject H? (evidence of a difference).
• Effect Size: A small p-value doesn’t mean the difference is practically significant (always interpret in context).

Quick Check Questions

1. Multiple Choice: A researcher tests whether a new fertilizer increases plant growth by measuring the height of 20 plants before and after treatment. Which test is appropriate?
2. (A) One-sample t-test
3. (B) Two-sample t-test
4. (C) Paired t-test

5. (D) Chi-square test for homogeneity Answer: (C) Paired t-test. The data is paired (same plants measured twice).

6. FRQ Part: A study records the blood pressure of 15 patients before and after a new medication. The mean difference (after – before) is d? = -5.2 mmHg with s_d = 8.1 mmHg.

7. a) State the hypotheses for a test to determine if the medication reduces blood pressure.
8. b) Are the conditions for inference met? Justify. Answer:
9. a) H?: ?_d = 0, H?: ?_d < 0 (one-tailed, since we’re testing for a reduction).

10. b) Conditions:

    • Paired: Same patients before/after-paired.
    • Independence: Patients are independent (assuming random selection).
    • Normality: n = 15 (small), so check histogram/NPP of differences (not shown here, but assume roughly normal).
    • Random: Assume data comes from a randomized experiment.

11. Multiple Choice: For a paired t-test with 25 pairs, what are the degrees of freedom?

12. (A) 23
13. (B) 24
14. (C) 25
15. (D) 48 Answer: (B) 24. df = n – 1 = 25 – 1 = 24.

Last-Minute Cram Sheet

1. Paired t-test = analyze differences (not raw data).
2. H?: ?_d = 0, H?: ?_d-0 (or >/< 0 for one-tailed).
3. df = n – 1 (where n = number of pairs).
4. Conditions: Paired, Independent pairs, Normal differences (check if n < 30), Random.
5. Test Statistic: ( t = \frac{\bar{d}}{s_d / \sqrt{n}} ).
6. TI-84: Enter differences in L1-T-Test (not 2-SampTTest!).
7. Confidence Interval: ( \bar{d} \pm t^* \left( \frac{s_d}{\sqrt{n}} \right) ).
8. If 0 is in the CI-fail to reject H? (no significant difference).
9. Don’t mix up paired vs. two-sample t-tests!
10. Always interpret conclusions in context (e.g., "There is evidence the medication reduces blood pressure").