Answer
Confidence interval: $37.7\lt x ̅\lt52.3$
Work Step by Step
We want to estimate the mean using a sample whose size is less than 30, but that was obtained from a population that is normally distributed.
$n=12$, so:
$d.f.=n-1=11$
$level~of~confidence=(1-α).100$%
$90$% $=(1-α).100$%
$0.9=1-α$
$α=0.1$
$t_{\frac{α}{2}}=t_{0.05}=1.796$ (According to Table VI, for d.f. = 11 and area in right tail = 0.05)
$Lower~bound=x ̅-t_{\frac{α}{2}}.\frac{s}{\sqrt n}=45-1.796\times\frac{14}{\sqrt {12}}=37.7$
$Upper~bound=x ̅+t_{\frac{α}{2}}.\frac{s}{\sqrt n}=45+1.796\times\frac{14}{\sqrt {12}}=52.3$
