Student-t Distribution

Context: FIT2086_MOC · the distribution the standardised sample mean follows once ${\sigma }^2$ is estimated rather than known · replaces $z$ with $t$ in the unknown-variance case of Confidence Intervals and of [[Tests for Normal Means (z-test and t-test)|the $t$-test]] · a wider-tailed sibling of the Gaussian Distribution

Quick Revision

• 🎯 Objective: $\frac{{\hat{\mu }}_{ML}-\mu }{{\hat{\sigma }}_u/\sqrt{n}}\sim t(n-1)$ ➔ one parameter, the degrees of freedom $\nu =n-1$; symmetric and self-similar like the normal, but with heavier tails.
• ⚡ Key Constraint: $\nu =n-1$, not $n$ — one degree of freedom is spent estimating $\mu$ inside ${\hat{\sigma }}_u^2$.

📝 Core

• Where it comes from ➔ substituting the estimate ${\hat{\sigma }}_u^2=\frac{1}{n-1}{\sum }_i(y_i-{\hat{\mu }}_{ML}{)}^2$ for the known ${\sigma }^2$ makes the standardised statistic no longer normal: the denominator is now random too.
• Extra uncertainty widens the tails ➔ $t(\nu )$ spreads probability further out and tails off to zero more slowly than $N(0,1)$; that surplus tail mass is the price of not knowing ${\sigma }^2$.
• Shape properties reused ➔ symmetric about $0$ and self-similar, so the same two-sided percentile trick as the unit normal works: the $100(1-\alpha /2)$-th percentile $t_{\alpha /2,\nu }$ cuts $\alpha /2$ off each tail.
• Limit ➔ as $\nu \to \infty$ the estimate ${\hat{\sigma }}_u^2$ concentrates on ${\sigma }^2$ and $t(\nu )\to N(0,1)$ ➔ $t_{\alpha /2,\nu }\downarrow z_{\alpha /2}$ from above, never below.
• Critical values are always $>$ the normal’s ➔ $t$-based intervals are wider than the corresponding $z$-based interval at the same $\alpha$, which is exactly how the coverage lost by estimating ${\sigma }^2$ is recovered.

📊 Two-Sided Critical Values ($\alpha =0.05$)

$n$	$\nu =n-1$	$t_{0.025,\nu }$	Excess over $z_{0.025}=1.96$
$3$	$2$	$\approx 4.30$	$+119\%$
$5$	$4$	$\approx 2.776$	$+42\%$
$6$	$5$	$\approx 2.57$	$+31\%$
$8$	$7$	$\approx 2.36$	$+20\%$
$10$	$9$	$\approx 2.262$	$+15\%$
$11$	$10$	$\approx 2.22$	$+13\%$
$15$	$14$	$\approx 2.145$	$+9\%$
$50$	$49$	$\approx 2.009$	$+2.5\%$
$100$	$99$	$\approx 1.984$	$+1.2\%$
$1000$	$999$	$\approx 1.962$	$+0.1\%$
$\infty$	$\infty$	$1.96$	$0\%$

Final extracted output: the penalty is severe only for tiny samples — by $n\approx 30$ the $t$ and $z$ intervals differ by a few percent, which is why large-$n$ work quotes $1.96$ without apology. Look-up syntax lives in R Toolkit (Cheatsheet); the coverage cost of ignoring the correction is measured empirically in Confidence Interval Coverage Simulation.

⚠️ Common Mistakes

• 💡 Using $\nu =n$ ➔ the degrees of freedom are $n-1$; at $n=8$ that is $t_{0.025,7}=2.36$, not $t_{0.025,8}=2.31$ — small, but it is a free mark.
• 💡 Feeding ${\hat{\sigma }}_{ML}^2$ into a $t$ interval ➔ the $t$ result is derived for the unbiased ${\hat{\sigma }}_u^2$ (divisor $n-1$); the ML version (divisor $n$) understates the spread (Estimator Quality (Bias, Variance, MSE)).
• 💡 Passing $\alpha$ where the percentile is wanted ➔ $t_{\alpha /2,\nu }$ is the $100(1-\alpha /2)$-th percentile, so the quantile argument is $p=1-\alpha /2=0.975$, not $0.025$ or $0.05$.

🧠 Active Recall

Why does estimating ${\sigma }^2$ force a different distribution rather than just a slightly noisier normal?

Answer

• Short answer: in $\frac{{\hat{\mu }}_{ML}-\mu }{{\hat{\sigma }}_u/\sqrt{n}}$ both numerator and denominator are random; the ratio of a normal to an independent random scale is no longer normal, and its exact law is $t(n-1)$.
• Why: Coverage, not cosmetics ➔ substituting ${\hat{\sigma }}_u$ into the $z$ interval yields less than $100(1-\alpha )\%$ coverage; the heavier $t$ tails are the correction that restores it, exactly when the population is normal.