Answer
Confidence interval: $262.4\lt x ̅\lt273.1$
We are 95% confident that the mean gestation period is between 262.4 and 273.1 days.
Work Step by Step
The population from which the data was extracted is normally distributed. Also, there are no outliers.
The mean:
$x ̅=\frac{266+261+270+260+277+270+278+269+258+252+275+277}{12}=267.75$
The standard deviation:
$s=\sqrt {\frac{(266-267.75)^2+(261-267.75)^2+(270-267.75)^2+(260-267.75)^2+(277-267.75)^2+(270-267.75)^2+(278-267.75)^2+(269-267.75)^2+(258-267.75)^2+(252-267.75)^2+(275-267.75)^2+(277-267.75)^2+}{12-1}}=8.49$
$n=12$, so:
$d.f.=n-1=11$
$level~of~confidence=(1-α).100$%
$95$% $=(1-α).100$%
$0.95=1-α$
$α=0.05$
$t_{\frac{α}{2}}=t_{0.025}=2.201$
(According to Table VI, for d.f. = 11 and area in right tail = 0.025)
$Lower~bound=x ̅-t_{\frac{α}{2}}.\frac{s}{\sqrt n}=267.75-2.201\times\frac{8.49}{\sqrt {12}}=262.4$
$Upper~bound=x ̅+t_{\frac{α}{2}}.\frac{s}{\sqrt n}=267.75+2.201\times\frac{8.49}{\sqrt {12}}=273.1$
