AP Statistics (AP Stats): Two?Sample t?test for Difference of Means

AP Statistics – Two?Sample t?test for Difference of Means

AP Statistics: Two-Sample t-test for Difference of Means

Exam-Ready Study Guide

What This Is

The two-sample t-test for the difference of means determines whether the average of one population differs from the average of another. It’s essential for comparing treatments (e.g., does a new fertilizer increase crop yield more than the old one?) or groups (e.g., do men and women spend different amounts on coffee per month?). The AP exam tests this in FRQs and MCQs, often requiring you to justify conditions, compute the test statistic, and interpret results in context.

Key Terms & Formulas

• Two-Sample t-test for – :
• H?: = (no difference in population means)
• H?: - (two-tailed), > (right-tailed), or < (left-tailed)
• Test statistic: t = (x – x) / ?(s?²/n? + s?²/n?)
  • x, x = sample means
  • s?, s? = sample standard deviations
  • n?, n? = sample sizes

• Degrees of freedom (df): Use min(n?–1, n?–1) or the conservative df (AP prefers this for simplicity). For exact df, use the Welch-Satterthwaite equation (not required on AP).

• Two-Sample t-Interval for – :

• Formula: (x – x) ± t* ?(s?²/n? + s?²/n?)

• t* = critical value from invT(area, df) (e.g., invT(0.975, df) for 95% CI).

• Conditions (LINER):

• Large samples or Normal populations: Check if n?-30 and n?-30 (CLT) or plot data (no skew/outliers).
• Independent samples: Groups must be independent (no pairing).
• Random: Data from random samples or randomized experiments.

• Equal variance (optional): If s?-s?, use pooled t-test (not required on AP).

• Calculator Commands (TI-84):

• 2-SampTTest: STAT-TESTS-4:2-SampTTest (for hypothesis test).
• 2-SampTInt: STAT-TESTS-0:2-SampTInt (for confidence interval).
• Input: Enter x?, s, n for both samples; choose Pooled: No (AP standard).

Step-by-Step / Process Flow

For a typical FRQ:
1. State hypotheses: - H?: = (no difference) - H?: - (or >/< depending on context).

1. Check conditions (LINER):
2. Normal/Large: n?-30 and n?-30 or show plots (boxplots/histograms).
3. Independent: Confirm samples are independent (e.g., separate groups).

4. Random: Data from random samples/experiments.

5. Compute test statistic:

6. Use formula or 2-SampTTest on TI-84.

7. Report t and df (use min(n?–1, n?–1)).

8. Find p-value:

9. Use tcdf(lower, upper, df) for two-tailed tests (e.g., tcdf(-1E99, -t, df) * 2).

10. For calculator: 2-SampTTest gives p-value directly.

11. Make conclusion:

12. Compare p-value to ? (e.g., 0.05).
13. Reject H? if p-value < ?-"There is convincing evidence that [context]."
14. Fail to reject H? if p-value-?-"There is not convincing evidence that [context]."

Common Mistakes

• Mistake: Using a paired t-test instead of two-sample.

• Correction: Paired tests require matched pairs (e.g., before/after measurements). Two-sample tests compare independent groups.

• Mistake: Forgetting to check normality for small samples.

• Correction: If n < 30, plot data (boxplot/histogram) to confirm no skew/outliers.

• Mistake: Mixing up – vs. – in hypotheses.

• Correction: Define groups clearly (e.g., " = treatment group, = control group").

• Mistake: Using pooled variance (AP does not require this).

• Correction: Always select Pooled: No in 2-SampTTest.

• Mistake: Misinterpreting the confidence interval.

• Correction: A CI for – that includes 0 means we fail to reject H? at that confidence level.

AP Exam Insights

• Tricky Distinction: The AP exam often tests two-sample t vs. paired t. Look for keywords like "independent groups" (two-sample) vs. "matched pairs" (paired).
• Common FRQ Setup: You’ll be given summary stats (x?, s, n) and asked to:
• State hypotheses.
• Check conditions.
• Compute a test statistic or CI.
• Interpret results in context.
• Calculator Pitfall: Forgetting to clear lists (L1, L2) before entering data for 2-SampTTest. Always check inputs!
• Degrees of Freedom: AP accepts conservative df (min(n?–1, n?–1)), but some FRQs may ask for it explicitly.

Quick Check Questions

1. MCQ: A researcher wants to test if the mean height of basketball players is greater than the mean height of soccer players. 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 Answer: (B) Two-sample t-test (comparing two independent groups).

6. FRQ Part: Two samples of students took a math test: Sample 1 (n?=40, x=82, s?=5) and Sample 2 (n?=50, x=79, s?=6). The 95% CI for – is (0.5, 5.5). Interpret this interval in context. Answer: We are 95% confident that the true mean math score for Sample 1 is between 0.5 and 5.5 points higher than for Sample 2.

7. MCQ: For a two-sample t-test, why is it important to check that the samples are independent?

8. (A) To ensure the test statistic follows a t-distribution.
9. (B) To avoid confounding variables.
10. (C) To satisfy the randomness condition.
11. (D) To justify using the pooled variance formula. Answer: (B) To avoid confounding variables (independence ensures groups don’t influence each other).

Last-Minute Cram Sheet

1. Hypotheses: H?: = ; H?: ?/ .
2. Test statistic: t = (x – x) / ?(s?²/n? + s?²/n?).
3. Conditions: LINER (Large/Normal, Independent, Random).
4. df: min(n?–1, n?–1) (conservative).
5. Calculator: 2-SampTTest (Pooled: No) or 2-SampTInt.
6. CI formula: (x – x) ± t* ?(s?²/n? + s?²/n?).
7. Always define and (e.g., " = treatment group").
8. For small samples (n < 30), check normality with plots.
9. A CI including 0-fail to reject H?.
10. Never use a two-sample t-test for paired data!