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7.6 Confidence Intervals for the Difference of Two Means

Part of Inference for Quantitative Data: Means in AP® Statistics. Read the opening section, then create a free account for the full interactive lesson, practice, and progress tracking.

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Two-Sample t Confidence Interval

Parameter: μ1−μ2{\mu_1 - \mu_2} = difference in two population means

Conditions:

  1. Random: Both samples are random (or randomized experiment)
  2. Independence: 10% condition for each group; groups independent of each other
  3. Normal / Large Sample: Each group — population normal OR ni≥30{n_i \geq 30}
Formula:
(xˉ1−xˉ2)±t∗s12n1+s22n2(\bar{x}_1 - \bar{x}_2) \pm t^* \sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}
Degrees of freedom: Use the conservative df=min⁡(n1−1,n2−1){df = \min(n_1 - 1, n_2 - 1)} by hand, or use the Welch-Satterthwaite approximation (calculator gives this):
df≈(s12/n1+s22/n2)2(s12/n1)2n1−1+(s22/n2)2n2−1df \approx \frac{(s_1^2/n_1 + s_2^2/n_2)^2}{\frac{(s_1^2/n_1)^2}{n_1-1} + \frac{(s_2^2/n_2)^2}{n_2-1}}

In this unit

9 lessons

  1. 7.1Introducing Statistics: Why Should I Worry About Error?Preview lesson
  2. 7.2Constructing a Confidence Interval for a Population MeanPreview lesson
  3. 7.3Justifying a Claim About a Population Mean Based on a Confidence IntervalPreview lesson
  4. 7.4Setting Up a Test for a Population MeanPreview lesson
  5. 7.5Carrying Out a Test for a Population MeanPreview lesson
  6. 7.6Confidence Intervals for the Difference of Two MeansThis lesson
  7. 7.7Justifying a Claim About the Difference of Two Means Based on a Confidence IntervalPreview lesson
  8. 7.8Setting Up a Test for the Difference of Two Population MeansPreview lesson
  9. 7.9Carrying Out a Test for the Difference of Two Population MeansPreview lesson

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