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Why Use a t-Statistic?

Two SigmaData Scientist · Technical Screen
HardData ScientistTechnical ScreenStatistics & Math

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This problem tests your conceptual understanding of the t-statistic, specifically how sample parameters influence its value and how it bridges the gap between estimation and hypothesis testing.

The Problem

You are working as a quant researcher analyzing the impact of a specific market signal on asset returns. You have collected a sample of returns X = {x_1, x_2, ..., x_n}.

  1. Define the t-statistic for testing the null hypothesis that the population mean return mu equals 0. Explicitly state the formula and define every variable used.

  2. Explain the intuitive meaning of the t-statistic: How does it treat the relationship between the observed mean, the noise (variance) in the data, and the sample size?

  3. Suppose you calculate a t-statistic of 2.1. If you were to increase your sample size n by a factor of 4 while keeping the sample mean and sample standard deviation constant, what happens to the t-statistic? Show your work.

  4. Discuss the "t-distribution" assumption. In what scenario would a researcher be justified in using a Z-statistic instead of a t-statistic, and why might that be dangerous in financial time series data?

Hints

Hint 1

Recall that the t-statistic is essentially a signal-to-noise ratio: the estimated mean divided by the standard error.

Hint 2

Consider how the standard error of the mean, `s / sqrt(n)`, scales as `n` increases.

Hint 3

Remember that the t-distribution approaches the standard normal distribution as the degrees of freedom (based on sample size) grow large.

Follow-up questions

  • How does the presence of autocorrelation (serial correlation) in your returns data invalidate the standard calculation of the t-statistic?
  • If your data is non-normally distributed with heavy tails, how does this affect the reliability of the p-value derived from the t-distribution?
  • In a machine learning context, how might you use bootstrapping to estimate the distribution of your mean effect if you are unsure if your data follows a specific distribution?