Answer to Question #247077 in Statistics and Probability for M_2

Question

Maximum breadth of samples of male Egyptian skulls from 4000 B.C. and 150 A.D.

, based on data from Ancient Races of the Thebaid by Thomson and Randall-Maciver:

4000 B.C. : 131 119 138 125 129 126 131 132 126 128 128 131

150 A.D. : 136 130 126 126 139 141 137 138 133 131 134 129

Changes in head sizes over time suggest interbreeding with people from other regions.

Find the sample standard deviation and use a 90% confidence interval to determine

whether the head sizes appear to have changed from 4000 B.C. to 150 A.D. Explain your

result.

Answer 1

Since there is no information about the population, t - test is the most appropriate in finding the confidence interval for the difference in the means of the two samples. We use the difference in the means because it provides a better judgement when comparing groups.

Therefore, the 90% confidence interval is given by; the formula;

$\bar{x1}-\bar{x2} \pm$ t$\alpha/2,v \sqrt{\frac{s1^2 }{n1} +\frac{s2^2}{n2}}$ , where $\bar{x1}$ ,$\bar{x2}$ ,$s1^2$ ,$s2^2$ are the means and variances of the two random samples.

V is the degrees of freedom and is given by;

$(\frac{s1^2}{n1} + \frac{s2^2}{n2})^2$ $/\frac{ (\frac{s1^2}{n1})^2}{n1-1} + \frac{(\frac{s2^2}{n2})^2}{n2-1}$

The degrees of freedom are calculated as above because we have unknown population variances that are assumed unequal.

From the data,

$\bar{x1}$ = 128.6667 $\bar{x2}$ = 133.3333

s1 = 4.63844 s2 = 5.015

n1 = 12 n2 = 12

$v =$ $(\frac{4.63844^2}{12} + \frac{5.015^2}{12})^2 / \frac{(\frac{4.63844^2}{12})^2}{11} +\frac{ (\frac{5.015^2}{12})^2}{11}$ = 21.8672

$\approxeq$ 22

t_{a/2, v} = t_{0.05 , 22} = 1.717

The 90% confidence interval is;

128.6667 - 133.3333 $\pm$ 1.717 $\sqrt{\frac{4.63844^2}{12} + \frac{5.015^2}{12}}$

= (-4.6666$\pm$ 3.3859)

= (-8.0525 , -1.2805)

Explanation of the results

From the confidence interval, we are 90% confident that the difference in the means of the two samples lies between -8.0525 and -1.2805.

Since the confidence interval does not include 0, we conclude that the head sizes of the male Egyptian skulls changed from 4000BC to 150AD,

and the difference is statistically significant