Consider again the Etruscan and Italian skull size data set. We have analyzed this from time to time. A complete two sample analysis is given at the end of this section. Based on this analysis there is a difference between Etruscan and Italian
skull sizes. Recall that scientists were trying to establish a link between
ancient Etruscans and modern Italians, (the Italian skulls were recent);
that is, the Etruscans were native to Italy. Our statistical analysis is not supportive of that link but does it really show that the Etruscans were not native to Italy? The problem here is that there are many other variables that could cause the change in skull size : diet, environmental changes, etc. There is no way to control these variables.

This is an observational study. These studies are important. This certainly is evidence against the link, but other evidence needs to be gathered. We'll come back to these discussions later, but first lets talk about controlled experimental designs.

Two sample analysis of Etruscan Italian example

Were the ancient Etruscan native to Italy? To help answer this question we have two samples of skull sizes. The first sample consists of the maximum head breadths of 84 Etruscan skulls while the second sample consists of the maximum head breadths of 70 modern Italian skulls. The data is given in Appendix A.

Comparison dotplot of the data sets: . : Etruscan :: ::: : .: : :: :::. :::: : . .: .:.: :.:: :::: :::: : :. :. : -------+---------+---------+---------+---------+---------C10 . Italian : :. : : .: :::: .::. :: . .. .: :::: :::: :::: ::: .. . -------+---------+---------+---------+---------+---------C10 120.0 128.0 136.0 144.0 152.0 160.0 Comparison boxplot of the data sets: ----------- Etruscan * -----------I + I------------ ----------- ----------- Italian * -----------I + I-------------- ----------- --------+---------+---------+---------+---------+--------C10 120.0 128.0 136.0 144.0 152.0The plots indicate that Etruscan skull sizes are larger than the Italian skull sizes. Furthermore, they indicate that it is a location problem.For formal inference, let Delta be the shift in location from typical Italian skull sizes to typical Etruscan skull sizes. We will first test the hypotheses:

versus

We will use the two sample Wilcoxon to test these hypotheses. (Just click on the class code: two sample wilcoxon). The Test statistics is

This far exceeds the expected value ofTunderH_{0}, which is 84(70)/2 = 2940. The p-value is .000. So we rejectH_{0}with high confidence. The Etruscan skull sizes are larger.The Wilcoxon estimate of shift in location is 11mm and the confidence interval for is (10, 13). So typical Etruscan head sizes are from 10 to 13mm larger than typical Italian skull sizes.