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Michael Brenndoerfer

Confidence Intervals and Test Assumptions › Choosing Between Z-Tests and T-Tests

Michael BrenndoerferUpdated January 12, 20263 min read

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Part 2 of 9Hypothesis Testing

Chapter 13Confidence Intervals and Test Assumptions › Choosing Between Z-Tests and T-Tests

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The choice between z-tests and t-tests depends on whether you know the population standard deviation.

The Z-Test: Known VarianceLink Copied

The z-test assumes you know the population standard deviation $\sigma$ :

z = \frac{\bar{x} - \mu_0}{\sigma / \sqrt{n}}

Under $H_0$ , this follows a standard normal distribution $N(0, 1)$ .

When do you know $\sigma$ ? Rarely. Possible scenarios include:

Measurement instruments with known precision from extensive calibration
Standardized tests with established population parameters
Very large historical datasets that effectively reveal the population

The T-Test: Unknown VarianceLink Copied

The t-test estimates $\sigma$ from the sample using $s$ :

t = \frac{\bar{x} - \mu_0}{s / \sqrt{n}}

Under $H_0$ , this follows a t-distribution with $n-1$ degrees of freedom.

Why the t-distribution? When you estimate $\sigma$ with $s$ , you introduce additional uncertainty. The sample standard deviation $s$ is itself a random variable. The t-distribution has heavier tails than the normal to account for this uncertainty.

The T-Distribution: Heavy Tails for Small SamplesLink Copied

Out[10]:

Visualization

Line plot comparing t-distributions with different degrees of freedom to the standard normal distribution. — The t-distribution has heavier tails than the normal distribution, especially for small degrees of freedom. This means extreme values are more likely under the t-distribution, which correctly accounts for the uncertainty from estimating σ with s. As df increases, the t-distribution approaches the normal.

The practical consequence: t-tests give larger p-values (and wider confidence intervals) than z-tests for the same data. This is appropriate because we're less certain when we don't know $\sigma$ .

Degrees of Freedom	t Critical Value (95% CI)	Compared to z = 1.96
5	2.571	31% larger
10	2.228	14% larger
20	2.086	6% larger
30	2.042	4% larger
100	1.984	1% larger
∞	1.960	Same as z

For practical purposes, with n > 30, the difference between t and z is usually negligible. But always use the t-test when $\sigma$ is unknown. There's no benefit to using z without a known $\sigma$ , and it can inflate your Type I error rate for small samples.