Answer
$30.9\lt\sigma^2\lt78.2$
$5.6\lt\sigma\lt8.8$
Work Step by Step
Given $n=27,s=6.8$, at a 90% confidence and $df=26$ the critical values are $\chi^2_{left}=15.379 $ and
$\chi^2_{right}=38.885 $, the range for the variance can then be found as
$\frac{(n-1)s^2}{\chi^2_{right}}\lt\sigma^2\lt \frac{(n-1)s^2}{\chi^2_{left}}$ which gives $30.9\lt\sigma^2\lt78.2$
and for the standard deviation, we have $5.6\lt\sigma\lt8.8$
