Small Samples & Student t

Large-sample normal inference and exact Student \(t\) inference rely on different arguments. Using a larger critical value cannot compensate for every failure of the model.

1. Exact one-sample Student \(t\) inference

Assume \(Y_{1},\ldots ,Y_{n}\) are \(i.i.d. \mathcal{N}(\mu ,\sigma ^{2})\), with \(\sigma ^{2}>0\) and \(n\ge 2\). Normal sampling gives three facts:

\[\begin{aligned}Z&=\frac{\sqrt n(\bar Y-\mu)}\sigma\sim\mathcal N(0,1),\\ W&=\frac{(n-1)s^2}{\sigma^2}\sim\chi^2_{n-1},\\ Z&\perp\!\!\!\perp W.\end{aligned}\]

These facts imply \((\bar{Y} -\mu )/(s/\sqrt{n})=Z/\sqrt{W/(n-1)}\sim t_{n-1}\). Under \(H_{0}\), replace \(\mu\) by \(\mu _{0}\). Independence and the chi-squared law are special normal-sample results, not consequences of unbiasedness of \(s^{2}\). The ratio derivation is supplied here; proving those normal-sample distribution facts belongs to the later distribution theory referenced by SW.

\[\operatorname{CI}_{1-\alpha}=\bar Y\pm t_{n-1,\,1-\alpha/2}\frac s{\sqrt n}.\]

For \(n=20\) the 95% critical value is about 2.093, instead of 1.96. A statistic 2.15 rejects at 5% under the exact normal-sample model. If the population is strongly nonnormal, neither the label “t-statistic” nor \(n=20\) justifies that exact result.

2. Exact pooled two-sample inference

Assume two independent samples, each i.i.d. normal, with respective means \(\mu _{A}, \mu _{B}\) and a common variance \(\sigma ^{2}\). Define:

\[\begin{aligned}s_p^2&=\frac{(n_A-1)s_A^2+(n_B-1)s_B^2}{n_A+n_B-2},\\ \operatorname{SE}_p&=s_p\sqrt{\frac1{n_A}+\frac1{n_B}},\\ t_p&=\frac{(\bar Y_A-\bar Y_B)-\delta_0}{\operatorname{SE}_p}\\&\sim t_{n_A+n_B-2}\quad\text{under }H_0.\end{aligned}\]

Each group'\(s\) deviations must be centered around its own group mean. The supplied PDF'\(s\) equation (3.23) prints the second group'\(s\) deviations around the first group'\(s\) mean; the notes use the mathematically correct within-group expression above and record this source error.

3. Equal sample sizes are not equal population variances

If \(n_{A}=n_{B}=m\), the pooled and unpooled standard errors are algebraically equal:

\[\begin{aligned}s_p^2&=\frac{s_A^2+s_B^2}{2},\\ \operatorname{SE}_p^2&=\frac{s_A^2+s_B^2}{m}=\operatorname{SE}_{\text{unpooled}}^2.\end{aligned}\]

But unequal population variances still generally prevent the pooled denominator from being the required common-scale chi-squared variable. Equal group sizes therefore do not alone imply an exact \(t_{2m-2}\) distribution. This distinction resolves potentially misleading wording in SW §3.6.

4. Unequal variances and Welch'\(s\) approximation

Optional extension beyond SW'\(s\) large-sample treatment

For independent normal groups with possibly unequal variances, the unpooled statistic is not generally exactly Student \(t\). Welch'\(s\) method approximates its degrees of freedom by:

\[\hat\nu=\frac{\left(s_A^2/n_A+s_B^2/n_B\right)^2}{\dfrac{(s_A^2/n_A)^2}{n_A-1}+\dfrac{(s_B^2/n_B)^2}{n_B-1}}.\]

This is an approximation, not a general exact distribution theorem. For Econ 141'\(s\) main large-sample route, use the unpooled SE and the justified normal approximation. Dependence, severe skewness, or rare outcomes require attention to the actual design and distribution; changing degrees of freedom alone is not a universal repair.

Setting	Reference distribution
One mean, i.i.d. normal, unknown variance	Exact \(t_{n-1}\)
One mean, general i.i.d. finite moments, large \(n\)	Asymptotic \(\mathcal{N}(0,1)\)
Two normal independent groups, equal variances, pooled SE	Exact \(t_{n§SUB8§+n§SUB9§-2}\)
Two independent groups, unequal variances, both large	Asymptotic \(\mathcal{N}(0,1)\), unpooled SE
Two normal groups, unequal variances, smaller samples	Welch approximation; generally not exact \(t\)

Check your understanding

Question. Both groups have 12 observations and very different variances. Does equal \(n\) establish an exact pooled \(t\) distribution?

Solution. No. It equates the two SE formulas, not the required distribution of the denominator.

Sources: SW §3.6, pp. 133–137; Exercise 3.21. Welch degrees of freedom are an optional supporting method. See the source log for equation (3.23) and its correction.