Answer
Confidence interval: $1.56\lt σ^2\lt2.77$
Work Step by Step
We want to estimate the population standard deviation using a sample obtained from a population that is normally distributed.
$n=25$. So:
$d.f.=n-1=24$
$level~of~confidence=(1-α).100$%
$95$% $=(1-α).100$%
$0.95=1-α$
$α=0.05$
$X_{1-\frac{α}{2}}^2=X_{0.975}^2=12.401$
(According to Table VII, for d.f. = 24 and area to the right of critical value = 0.975)
$X_{\frac{α}{2}}^2=X_{0.025}^2=39.364$
(According to Table VII, for d.f. = 24 and area to the right of critical value = 0.025)
$Lower~bound=\sqrt{\frac{(n-1)s^2}{X_{\frac{α}{2}}^2}}=\sqrt{\frac{24\times3.97}{39.364}}=\sqrt{2.420}=1.56$
$Upper~bound=\sqrt{\frac{(n-1)s^2}{X_{1-\frac{α}{2}}^2}}=\sqrt{\frac{24\times3.97}{12.401}}=\sqrt{7.683}=2.77$
