Hypothesis Testing with Two Samples

Background: Two-Sample t-Tests

Two-sample t-tests compare population means using two independent or paired samples.

1. Independent t-test: Compares means of two unrelated groups (e.g., measurements from two sites).
   • Assumptions:
     • Independence between groups
       
     • Normality (or large sample size for CLT)
       
     • Homogeneity of variances not required (default in R uses Welch’s correction).
   • Test Statistic:
     \[ t = \frac{\bar{x}_1 - \bar{x}_2}{\sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}} \]
     Degrees of freedom approximated via Welch-Satterthwaite equation.
2. Paired t-test: Compares means of two related groups (e.g., pre/post measurements).
   • Assumptions:
     • Paired observations
       
     • Differences approximately normal
   • Test Statistic:
     \[ t = \frac{\bar{d}}{s_d / \sqrt{n}} \]
     where \(\bar{d}\) = mean difference, \(s_d\) = standard deviation of differences.

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Example 1: Independent t-Test

Using the built‐in mtcars dataset to compare miles per gallon between automatic (0) and manual (1) transmissions.

Example 2: Paired t-Test

Use the built‐in sleep dataset to compare extra sleep under two drug treatments.