1. The Hypothesis Testing Framework

In classical frequentist inference, we test a Null Hypothesis \(H_0\) against an Alternative Hypothesis \(H_1\):

• P-Value: The probability of observing a test statistic at least as extreme as the sample outcome, assuming \(H_0\) is strictly true: \(P(\text{Data} \mid H_0)\).
• Significance Level (\(\alpha\)): Typically \(\alpha = 0.05\). If \(p < \alpha\), we reject \(H_0\).

2. Two-Sample Student's t-Test & Z-Score

\[ t = \frac{\bar{x}_1 - \bar{x}_2}{\sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}} \]

3. Errors in Statistical Decision Making

• Type I Error (\(\alpha\)): False Positive — rejecting \(H_0\) when \(H_0\) is actually true.
• Type II Error (\(\beta\)): False Negative — failing to reject \(H_0\) when \(H_1\) is true. Statistical Power is \(1 - \beta\).