Biostatistics Homework 4

This homework asks you to think about your next mini-project: Mini-project 3

Problem: Is T-Rex warm-blooded or cold-blooded? The data you will analyze reflect oxygen isotope compositions in various bones in one well-preserved Tyrannosaurus Rex skeleton found in Montana. The data were read from a graph in the Science article by Barrick and Showers (1994).

The question is whether this T-rex seems to be warm-blooded or cold-blooded. That is, were the bones in the core formed at roughly the same temperature as the bones in the extremities or is there a substantial (>4 degrees C) differential in the temperatures at which the bones were formed? There are, of course, other possibilities besides being warm-blooded or cold-blooded, although the debate is often phrased using those terms. The only thing one might be able to tell is whether it seems likely that T-Rex used some kind of thermal regulation to keep his body temperature nearly constant.

The isotope composition numbers do not tell us the precise temperatures at which the bones were formed but can tell us the temperature differential. If bone(s) in group 1 have an average isotope composition of X and bones in group 2 have an average isotope composition of Y then the temerature difference, in degrees Celsius, between the two groups is 4.2(Y-X). (Lower isotope composition levels correspond to higher body temperatures, by the way.) Similarly, if the confidence interval for the difference in isotope composition in two groups is (.5, 1.1) say, then the confidence interval for the temperature difference, in degrees Celsius, is 4.2*(.5, 1.1) = (2.1, 4.62). Similarly, a test of whether there is a temperature difference of at least 4 degrees Celsius is a test of whether there is an isotope composition difference of at least 4/4.2=0.952381.

1. Describe, mathematically, how you test whether the difference between two populations, X and Y, is at least 0.95. Normally we test whether the difference is positive, negative, or non-zero.

2. Part of Barrick's and Showers's argument that T-Rex was warm-blooded was the following: a one-sided 2-sample t-test of the warmest core bone (Dorsal vertebra 1) and the coldest extermal bone (Distal caudal) for whether the difference is at least .952381 yields a t-test statistic of 0.66 with a p-value of 0.27. Thus there is no evidence against temperatures remaining within 4 degrees between bones and thus the claim that T-Rex had a method for regulating his body temperature is supported.

   You do not have to replicate this argument with the data - that you will do in the mini-project next week. For now, just assess the merit of the argument those authors made. Is it a sound argument or are there flaws in it?