t or z?

Question

When do you use a t-test instead of a z-test, what exactly changes in the mechanics, when does the distinction stop mattering, and what assumption do both tests make that market data routinely violates?

Accepted Answer

The rule. Use z when the population standard deviation is known (or n is large enough that estimating it is free); use t when sigma is estimated by the sample s and n is small. Replacing sigma with a noisy s adds estimation error to the denominator of bar x / (s/sqrt n), fattening its tails: the statistic follows Student's t with n-1 degrees of freedom rather than a normal. What changes numerically: critical values for a two-sided 5% test are 2.776 at textdf=4, 2.262 at textdf=9, 2.042 at textdf=30, converging to 1.96. By n approx 50-100 the difference is cosmetic. The t-distribution is the price of not knowing sigma, paid in wider intervals. Fine print: the exact t distribution assumes the underlying data is normal. For non-normal data the CLT still rescues the numerator at large n, but fat tails slow the convergence (the kurtosis shows up in the variance of s^2), so small-sample inference on heavy-tailed P&L is shakier than either table admits. What both assume and markets violate: independence. Autocorrelated observations, overlapping returns, persistent vol, smoothed marks, make the standard error itself wrong, typically understated, regardless of which table you read at the en

t or z?

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