Answer
Confidence interval: $52.34\lt x ̅\lt57.26$
As the sample size increases, the width of the interval decreases.
Work Step by Step
$n=51$, so:
$d.f.=n-1=50$
$level~of~confidence=(1-α).100$%
$90$% $=(1-α).100$%
$0.9=1-α$
$α=0.1$
$t_{\frac{α}{2}}=t_{0.05}=1.676$
(According to Table VI, for d.f. = 50 and area in right tail = 0.05)
$Lower~bound=x ̅-t_{\frac{α}{2}}.\frac{s}{\sqrt n}=54.8-1.676\times\frac{10.5}{\sqrt {51}}=52.34$
$Upper~bound=x ̅+t_{\frac{α}{2}}.\frac{s}{\sqrt n}=54.8+1.676\times\frac{10.5}{\sqrt {51}}=57.26$
