Answer
Confidence interval: $861.98\lt x ̅\lt864.02$
We are 99% confident that the mean muzzle velocity is between 861.98 and 864.02 meters per second.
Work Step by Step
We want to estimate the mean using a sample whose size is greater than 30.
$n=50$, so:
$d.f.=n-1=49$
$level~of~confidence=(1-α).100$%
$99$% $=(1-α).100$%
$0.99=1-α$
$α=0.01$
$t_{\frac{α}{2}}=t_{0.005}=2.678$
(According to Table VI, for d.f. = 50, the closest value to 49, and area in right tail = 0.005)
$Lower~bound=x ̅-t_{\frac{α}{2}}.\frac{s}{\sqrt n}=863-2.678\times\frac{2.7}{\sqrt {50}}=861.98$
$Upper~bound=x ̅+t_{\frac{α}{2}}.\frac{s}{\sqrt n}=863+2.678\times\frac{2.7}{\sqrt {50}}=864.02$
