Answer
Confidence interval: $51.4\lt x ̅2\lt56.6$
We are 95% confident that the mean length of sentencing is between 51.4 and 56.6 months.
Work Step by Step
We want to estimate the mean using a sample whose size is greater than 30.
$n=40$, so:
$d.f.=n-1=39$
$level~of~confidence=(1-α).100$%
$95$% $=(1-α).100$%
$0.95=1-α$
$α=0.05$
$t_{\frac{α}{2}}=t_{0.025}=2.023$
(According to Table VI, for d.f. = 39 and area in right tail = 0.025)
$Lower~bound=x ̅-t_{\frac{α}{2}}.\frac{s}{\sqrt n}=54-2.023\times\frac{8}{\sqrt {40}}=51.4$
$Upper~bound=x ̅+t_{\frac{α}{2}}.\frac{s}{\sqrt n}=54+2.023\times\frac{8}{\sqrt {40}}=56.6$
