## Biostatistics Mini-project 3

The following data reflect oxygen isotope compositions in various bones in one well-preserved Tyrannosaurus Rex skeleton found in Montana. The data are read from a graph in the Science article by Barrick and Showers (1994). They report data from 54 measurements in 12 bones and it is difficult to discern more than 52 measurements from their graph. You should simply be aware that these data are not exactly their data but are as close to their data as I can estimate.

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.

You can answer these questions from the isotope data using the following fact: 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.

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 giving no evidence against temperatures remaining within 4 degrees in each bone and thus supporting the claim that T-Rex had a method for regulating his body temperature. Figure out how to replicate these results and assess this argument. Also make your own argument about whether these data support the hypothesis that T-Rex was warm-blooded, that he was cold-blooded, or whether the data are inconclusive on the matter.

```region	bone	oxygen_isotope_composition
core	Dorsal_vertebra_1	10.61
core	Dorsal_vertebra_1	10.90
core	Dorsal_vertebra_1	11.15
core	Dorsal_vertebra_1	11.29
core	Dorsal_vertebra_1	11.43
core	Rib_16	11.10
core	Rib_16	11.23
core	Rib_16	11.25
core	Rib_16	11.28
core	Rib_16	11.49
core	Gastralia_1	11.34
core	Gastralia_1	11.38
core	Gastralia_1	11.70
core	Gastralia_2	11.59
core	Gastralia_2	11.78
core	Gastralia_2	12.04
core	Dorsal_vertebra_2	10.93
core	Dorsal_vertebra_2	11.22
core	Dorsal_vertebra_2	11.33
core	Dorsal_vertebra_2	11.64
core	Dorsal_vertebra_2	11.70
mid	Femur	11.68
mid	Femur	11.80
mid	Femur	11.87
mid	Femur	12.14
mid	Proximal_caudal	10.90
mid	Proximal_caudal	11.01
mid	Proximal_caudal	11.06
mid	Proximal_caudal	11.11
mid	Proximal_caudal	11.27
mid	Proximal_caudal	11.34
extremity	Tibia	11.30
extremity	Tibia	11.39
extremity	Tibia	11.64
extremity	Tibia	12.13
extremity	Tibia	12.27
extremity	Metatarsal	11.30
extremity	Metatarsal	11.64
extremity	Metatarsal	11.94
extremity	Metatarsal	12.14
extremity	Phalange	11.55
extremity	Phalange	11.87
extremity	Phalange	12.03
extremity	Mid_caudal	11.33
extremity	Mid_caudal	11.40
extremity	Mid_caudal	11.52
extremity	Mid_caudal	11.56
extremity	Mid_caudal	11.90
extremity	Distal_caudal	11.94
extremity	Distal_caudal	11.97
extremity	Distal_caudal	12.02
extremity	Distal_caudal	12.24
extremity	Distal_caudal	12.26
extremity	Distal_caudal	12.39
```