Hypothesis Testing

Section: Hypothesis Testing and Statistical Inference Estimated study time: 45 minutes Content: Hypothesis testing is the formal framework for using sample data to make inferences about a population. In investment analysis, it is used to determine whether a strategy generates statistically significant alpha, whether two portfolio means differ, or whether a regression coefficient is meaningfully different from zero. The process begins with stating two competing hypotheses: the null hypothesis (H0) is the presumption of no effect (e.g., mean return = 0), and the alternative hypothesis (Ha) is what the analyst seeks to establish (e.g., mean return > 0). The alternative can be one-tailed (directional: > or <) or two-tailed (non-directional: ≠). A one-tailed test is used when theory or prior evidence strongly supports a specific direction; a two-tailed test is more conservative and appropriate when the direction is uncertain. The test statistic is calculated from sample data and compared to a critical value from the appropriate distribution to decide whether to reject H0. For tests of population means with known variance, the z-statistic is used: z = (X_bar – μ0) / (σ / √n). For unknown variance (the common case), the t-statistic with n–1 degrees of freedom is used: t = (X_bar – μ0) / (s / √n). The p-value is the probability of obtaining a test statistic at least as extreme as the one observed, assuming H0 is true. If p-value < significance level (α), we reject H0. Commonly used significance levels are 1%, 5%, and 10%, corresponding to 99%, 95%, and 90% confidence levels. The significance level α is the probability of making a Type I error — rejecting H0 when it is actually true (a false…