Question

The Etruscans. Anthropologists are still trying to unravel the mystery of the origins of the Etruscan empire, a highly advanced Italic civilization formed around the eighth century B.C. in central Italy. Were they native to the Italian peninsula or, as many aspects of their civilization suggest, did they migrate from the East by land or sea? The maximum head breadth, in millimeters, of 70 modern Italian male skulls and 84 preserved Etruscan male skulls was analyzed to help researchers decide whether the Etruscans were native to Italy. The resulting data can be found on the WeissStats CD. [SOURCE: N. Barnicot and D. Brothwell. "The Evaluation of Metrical Data in the Comparison of Ancient and Modern Bones." In Medical Biology and Etruscan Origins, G. Wolstenholme and C. O' Connor, cds., Littlc, Brown \& Co., 1959?

(a) Obtain normal probability plots, boxplots, and the standard deviations for the two samples.
(b) Based on your results from part (a). which would you be inclined to use to compare the population means: a pooled or a nonpooled tt-procedure? Explain your answer.
(c) Do the data provide sufficient evidence to conclude that a difference exists between the mean maximum head breadths of modern Italian males and Etruscan males? Perform the required hypothesis test at the 5% significance level by using both the pooled t-test and the nonpooled t-test. Compare your results.
(d) Find a 95% confidence interval for the difference between the mean maximum head breadths of modern Italian males and Etruscan males. Obtain the required confidence interval by using both the pooled t-interval procedure and the nonpooled t-interval procedure. Compare your results.

Short answer

(a) The normal probability plots is given by:

The boxplot for the given data is:

(b) From the probability plots and boxplots, we can conclude that there is one outlier for both the maximum head breaths for modern Italian males and Etruscan males, and some data points are far from the straight line and one data point lies outside the control limits for probability plot.

Therefore, the graph reveals, the nonpoold t-test is moderately assumption of normality, therefore assumption 3 is wrong.

Hence it is not satisfied, we cannot do the parts (c)-(d).

Step-by-step solution

Part (a) Step 1. GIven Information.  

The maximum head breadth, in millimeters, of 70 modern Italian male skulls and 84 preserved Etruscan male skulls was analyzed to help researchers decide whether the Etruscans were native to Italy. The resulting data can be found on the WeissStats CD.

Part (a) Step 2. Summary statement. 

The summary statistics for two samples of the maximum head breaths for modern Italian males and Etruscan males is:

Part (a) Step 3. Normal probability plot. 

Using MINITAB output, the normal probability is:

Part (a) Step 4. BOX plot for the treatment. 

The boxplot by using MINITAB is:

Part (b) Step 1. A  pooled or a nonpooled t-procedure  

Fromm the summary statistics for the two samples, the maximum head breaths for modern Italian males and Etruscan males is are approximately same. So the pooled t-test is appropriate than nonpooled test.

To check three conditions for pooled t-test:

The data obtained from a randomized comparative experiment. Therefore, assumptions 1 and 2are satisfied.

From the probability plots and boxplots, we can conclude that there is one outlier for both the maximum head breaths for modern Italian males and Etruscan males, and some data points are far from the straight line and one data point lies outside the control limits for probability plot.

Therefore, the graph reveals, the nonpoold t-test is moderately assumption of normality, therefore assumption 3 is wrong.

Hence it is not satisfied, we cannot do the parts (c)-(d).