# How do you adjust a two-sample t-test to accomodate samples of different sizes?

Mar 2, 2016

You adjust the statistic and the degrees of freedom for the differing sample sizes.

#### Explanation:

The two sample t-test can be defined to accommodate different sample sizes and different sample variances in the most general case. The statistic is defined as the function of the two random variables, each variable being the sample mean of each sample:

$T = \frac{{m}_{1} - {m}_{2}}{\sqrt{{s}_{1}^{2} / {N}_{1} + {s}_{2}^{2} / {N}_{2}}}$

The degrees of freedom needs to be approximated by the following formula:

$\nu = {\left({s}_{1}^{2} / {N}_{1} + {s}_{2}^{2} / {N}_{2}\right)}^{2} / \left({\left({s}_{1}^{2} / {N}_{1}\right)}^{2} / \left({N}_{1} - 1\right) + {\left({s}_{2}^{2} / {N}_{2}\right)}^{2} / \left({N}_{2} - 1\right)\right)$

You then proceed with the normal t-test by comparing our statistic, $T$, to the critical value of the t-distribution, ${t}_{p , \nu}$.

This information was taken from the following link:
http://www.itl.nist.gov/div898/handbook/eda/section3/eda353.htm