The log return $X$ on a certain stock investment is an $N(\mu,\sigma^2)$ random variable.
A financial analyst has claimed that the volatility $\sigma$ of the log return on this stock is less than $3$ units. A random sample of $11$ returns on this stock gave an estimated variance of the log-returns as $s^2 = 16$.
Assess the analyst's claim by using a significance test at level $\alpha = 0.05$ to test
$H_0: \sigma^2 \leq 9 \text{ against } H_1: \sigma^2 > 9.$
Find a two-sided $95\%$ confidence interval for $\sigma^2$.