CFA Level I: Hypothesis Testing

Fundamentals of Hypothesis Testing

Hypothesis testing is a statistical method used to evaluate investment claims rigorously. It involves stating a null hypothesis $H_0$ and an alternative hypothesis $H_1$, then using sample data to determine whether to reject $H_0$.

Key Statistical Concepts

The test statistic is calculated from sample data and compared to a critical value or p-value. The significance level $\alpha$ is the probability of rejecting $H_0$ when it is true (Type I error). Type II error $\beta$ is the probability of failing to reject $H_0$ when it is false. The power of a test is $1-\beta$.

Common Hypothesis Tests

The t-test is used for testing population means and mean differences. The chi-square test is for variance, and the F-test is for equality of variances. For example, the t-statistic for a sample mean is calculated as $t=\frac{\bar{x}-{\mu }_0}{s/\sqrt{n}}$. These tests are applied in investment manager performance evaluation to test claims about returns or risk.

Parametric vs. Nonparametric Tests

Parametric tests assume a specific distribution (e.g., normal), while nonparametric tests do not. In finance, parametric tests are common, but nonparametric tests are used when assumptions are violated.

Common Pitfalls

Common pitfalls in hypothesis testing include misinterpreting p-values, ignoring Type II errors, and using inappropriate tests for the data. It is also critical to check assumptions such as normality and independence.

Summary

• Hypothesis testing provides a framework for rigorously evaluating investment claims.
• Key concepts include test statistics, significance levels, Type I and Type II errors.
• Common tests include t-tests for means, chi-square for variance, and F-tests for variance equality.
• Parametric tests assume underlying distributions, while nonparametric tests are distribution-free.
• Hypothesis testing is applied in investment manager performance evaluation.