archaeologists have studied sizes of egyptian skulls in an attempt to determine whether breeding occurred…

Explanation:

Step1: Calculate sample mean

First, find the sum of the data values: $127.9+138.1+126.2+131.9+142.9+136.1+128+129.3 = 1050.4$. There are $n = 8$ data - points. The sample mean $\bar{x}=\frac{1050.4}{8}=131.3$.

Step2: Calculate sample standard deviation

The formula for the sample standard deviation $s=\sqrt{\frac{\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}}{n - 1}}$. $x_1-\bar{x}=127.9 - 131.3=-3.4$, $(x_1 - \bar{x})^2 = 11.56$; $x_2-\bar{x}=138.1 - 131.3 = 6.8$, $(x_2 - \bar{x})^2=46.24$; $x_3-\bar{x}=126.2 - 131.3=-5.1$, $(x_3 - \bar{x})^2 = 26.01$; $x_4-\bar{x}=131.9 - 131.3 = 0.6$, $(x_4 - \bar{x})^2=0.36$; $x_5-\bar{x}=142.9 - 131.3 = 11.6$, $(x_5 - \bar{x})^2 = 134.56$; $x_6-\bar{x}=136.1 - 131.3 = 4.8$, $(x_6 - \bar{x})^2=23.04$; $x_7-\bar{x}=128 - 131.3=-3.3$, $(x_7 - \bar{x})^2 = 10.89$; $x_8-\bar{x}=129.3 - 131.3=-2$, $(x_8 - \bar{x})^2 = 4$. $\sum_{i = 1}^{8}(x_{i}-\bar{x})^{2}=11.56 + 46.24+26.01+0.36+134.56+23.04+10.89+4=256.66$. $s=\sqrt{\frac{256.66}{7}}\approx6.04$.

Step3: Determine the t - value

For a 99% confidence interval with $n - 1=7$ degrees of freedom, the $t$-value $t_{\alpha/2}$ from the t - distribution table is $t_{0.005,7}=3.499$.

Step4: Calculate the margin of error

The margin of error $E=t_{\alpha/2}\frac{s}{\sqrt{n}}=3.499\times\frac{6.04}{\sqrt{8}}\approx3.499\times2.137\approx7.47$.

Step5: Calculate the confidence interval

The lower limit is $\bar{x}-E=131.3 - 7.47 = 123.83$. The upper limit is $\bar{x}+E=131.3 + 7.47 = 138.77$.

Answer:

$123.83<\mu<138.77$