Answer
Confidence interval: $51.54\lt x ̅\lt58.06$
Work Step by Step
$n=30$, so:
$d.f.=n-1=29$
$level~of~confidence=(1-α).100$%
$90$% $=(1-α).100$%
$0.9=1-α$
$α=0.1$
$t_{\frac{α}{2}}=t_{0.05}=1.699$
(According to Table VI, for d.f. = 29 and area in right tail = 0.05)
$Lower~bound=x ̅-t_{\frac{α}{2}}.\frac{s}{\sqrt n}=54.8-1.699\times\frac{10.5}{\sqrt {30}}=51.54$
$Upper~bound=x ̅+t_{\frac{α}{2}}.\frac{s}{\sqrt n}=54.8+1.699\times\frac{10.5}{\sqrt {30}}=58.06$
