T-Tests and Unknown Variance Enrichment

Scope Label

Enrichment / outside current confirmed Core 9758 path. This branch is retained for broader statistical understanding. The core revision path for this wiki is the $Z$-test for a single population mean with known population variance.

Why Unknown Variance Changes the Problem

This enrichment branch should not change the main revision route. Learn the core $Z$-test first. Then use this branch to understand why a different distribution appears when the population variance is not known.

In the core $Z$-test, the population standard deviation $\sigma$ is known, so the test statistic is

$$Z=\frac{\bar{X}-{\mu }_0}{\sigma /\sqrt{n}}.$$

If $\sigma$ is unknown, it is common to replace it with the sample standard deviation $s$:

$$\frac{\bar{X}-{\mu }_0}{s/\sqrt{n}}.$$

But this replacement introduces extra uncertainty because $s$ is itself computed from the sample.

Choosing Between $Z$ and $t$

The broad decision logic is:

• if ${\sigma }^2$ is known, use a $Z$-test;
• if ${\sigma }^2$ is unknown and $n$ is large, a large-sample normal approximation may be used;
• if ${\sigma }^2$ is unknown, $n$ is small, and the population is normal or assumed normal, a $t$-test may be used;
• if ${\sigma }^2$ is unknown, $n$ is small, and normality cannot be assumed, the usual small-sample mean test is not justified in this framework.

Variance information	Sample size / population condition	Test idea
${\sigma }^2$ known	population normal, any $n$	core $Z$-test
${\sigma }^2$ known	$n$ large	core large-sample $Z$-test
${\sigma }^2$ unknown	$n$ large	enrichment large-sample approximation using $s$
${\sigma }^2$ unknown	small sample, normal population assumed	enrichment $t$-test
${\sigma }^2$ unknown	small sample, no normality assumption	not justified in this framework

How to read this enrichment figure. Known $\sigma$ leads to the core $Z$ route. Replacing $\sigma$ by sample $s$ introduces extra uncertainty; the small-sample normal-population route then uses a $t$ distribution.

The $t$-Distribution

When $\sigma$ is replaced by $s$, the statistic has more uncertainty than a standard normal statistic.

For a random independent sample from a normal population, define $s$ as the sample standard deviation obtained from the unbiased sample variance

$$s^2=\frac{1}{n-1}\sum _{i=1}^n(X_i-\bar{X}{)}^2.$$

Then the statistic

$$T=\frac{\bar{X}-{\mu }_0}{s/\sqrt{n}}$$

follows a $t$-distribution with $n-1$ degrees of freedom under $H_0$.

The $t$-distribution:

• is centred at $0$;
• is symmetric;
• has heavier tails than $N(0,1)$;
• becomes closer to $N(0,1)$ as the degrees of freedom increase.

How to read this enrichment figure. Both curves are symmetric about zero, but the $t$ curve has lower centre and heavier tails. As degrees of freedom increase, it approaches the standard normal curve.

Enrichment Example: Small-Sample $t$-Test

A sample of size $6$ is taken from a population assumed to be normal. The sample mean is $12.4$ and the sample standard deviation is $3.0$.

Test at the $5\%$ level whether the population mean is greater than $10$.

Let $\mu$ be the true population mean.

$$H_0:\mu =10,H_1:\mu >10.$$

Since $\sigma$ is unknown, $n=6$ is small, and normality is assumed, use

$$T=\frac{\bar{X}-{\mu }_0}{s/\sqrt{n}},T\sim t_5\text{ under }{H}_0.$$

The observed statistic is

$$t_{\text{obs}}=\frac{12.4-10}{3/\sqrt{6}}\approx \mathrm{1.96.}$$

For an upper-tailed $5\%$ test with $5$ degrees of freedom, the critical value is approximately $2.015$.

Since

$$1.96<2.015,$$

do not reject $H_0$.

Conclusion: there is insufficient evidence at the $5\%$ level that the population mean is greater than $10$.

Enrichment: Large-Sample Unknown-Variance Test

When $n$ is large and $\sigma$ is unknown, replacing $\sigma$ by $s$ is often handled using a large-sample normal approximation:

$$Z\approx \frac{\bar{X}-{\mu }_0}{s/\sqrt{n}}.$$

This is not the same situation as the core known-variance $Z$-test. The approximation relies on large sample size.

Keep the distinction clear:

• known $\sigma$: standard core $Z$-test;
• unknown $\sigma$, small normal sample: $t$-test enrichment;
• unknown $\sigma$, large sample: large-sample approximation enrichment.

Common Pitfalls

• Using a $t$-test just because the sample size is small, without checking unknown variance and normality assumptions.
• Using the known-$\sigma$ $Z$-test when $\sigma$ is not given.
• Treating $s$ as if it were the true population standard deviation.
• Forgetting that the degrees of freedom are $n-1$.
• Mixing this enrichment branch into the current core revision path.

Revision Checklist

• Can you explain why replacing $\sigma$ by $s$ adds uncertainty?
• Can you state when a small-sample $t$-test is appropriate?
• Can you identify the degrees of freedom?
• Can you explain why the $t$-distribution has heavier tails?
• Can you keep this enrichment path separate from the core $Z$-test path?