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TL;DR

A test for comparing means when the population variance is unknown. The test statistic \( t = \frac{\bar{x} - \mu_0}{s/\sqrt{n}} \) follows a t-distribution with \( n-1 \) degrees of freedom under the null.

By Valenke Exam Prep Team·Last updated 2026-06-03

Student's t-Test

A test for comparing means when the population variance is unknown. The test statistic \( t = \frac{\bar{x} - \mu_0}{s/\sqrt{n}} \) follows a t-distribution with \( n-1 \) degrees of freedom under the null.

Why it matters for interviews

The t-test is used to determine if a strategy's mean return is significantly different from zero, if two strategies differ, or if a regression coefficient is significant. It is the most common statistical test in finance.

Definition and Mathematical Foundation

A test for comparing means when the population variance is unknown. The test statistic \( t = \frac{\bar{x} - \mu_0}{s/\sqrt{n}} \) follows a t-distribution with \( n-1 \) degrees of freedom under the null.

Application in Quantitative Finance

The t-test is used to determine if a strategy's mean return is significantly different from zero, if two strategies differ, or if a regression coefficient is significant. It is the most common statistical test in finance.

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Frequently Asked Questions

Why use the t-distribution instead of the normal?▼
The t-distribution accounts for the additional uncertainty from estimating the variance. With small samples, it has heavier tails than the normal, producing wider confidence intervals. As \( n \to \infty \), it converges to the normal.
What is Welch's t-test?▼
A modification for comparing two groups with unequal variances. It adjusts the degrees of freedom using the Welch-Satterthwaite approximation. It is more robust than the equal-variance t-test.
How many observations do you need for a reliable t-test on returns?▼
The standard error of the Sharpe ratio is approximately \( 1/\sqrt{n} \). To detect a Sharpe of 1.0 at 5% significance, you need about 4 years of monthly data (n=48). For Sharpe 0.5, about 16 years.