Answer
The confidence intervals overlap, so we cannot say that the lengths of skull breadths have changed over the time.
Work Step by Step
Since we know that $\alpha $ is .05, we know that $ t_{\alpha/2} $ is 2.201. In addition, using a standard deviation calculator, we find that the standard deviation of the 150 AD data set is 5.02. Thus, it follows that the error is:
$=\frac{2.201\times5.02}{\sqrt{n}}=\frac{2.201\times5.02}{\sqrt{12}}=3.1896$
We do the same thing for the 4000 BC data set.
Since we know that $\alpha $ is .05, we know that $ t_{\alpha/2} $ is 2.201. In addition, using a standard deviation calculator, we find that the standard deviation of the 4000 BC data set is 4.64. Thus, it follows that the error is:
$=\frac{2.201\times5.02}{\sqrt{n}}=\frac{2.201\times4.64}{\sqrt{12}}=3.464$
Thus, using the means as the center of the data set and the errors to determine the minimums and maximums, we see that the data overlap, meaning that we cannot say that the lengths of skull breadths have changed over the time.
