Importance of estimating $\sigma^2$ in linear Statistical model

Question

Statistical model for Complete Randomized design

$y_{ij} = \mu + \tau_i + \epsilon_{ij}$

where, $i$ denotes treatment and $j$ denotes observation.

$i=1,2,...,k\quad and \quad j=1,2,..., n_i$

$y_{ij}$ be a random variable that represents the response obtained on the $jth$ observation of the $ith$ treatment.

$\mu$ is the overall mean of the response $y_{ij}$

$\tau_i$ is the effect on the response of $ith$ treatment.

$\mu_i = \mu + \tau_i$

here $\mu_i$ denotes the true response of the $ith$ treatment.

and $\epsilon_{ij}$ is the random error term represent the sources of nuisance variation that is, variation due to factors other than treatments.

the assumption is $\epsilon_{ij}\sim^{iid} N(0,\sigma^2)$

By Least Square Estimation Procedure we estimate $\mu\quad and\quad\tau_i$.

• Why is it also important to estimate $\sigma^2$? If we do not estimate it , what will be the effect?

Any help including reference will be appreciated.

Answer 1

Without estimating the error variance you cannot estimate the variability in your parameters and hence cannot conduct hypothesis tests.