Locating, restoring and recompiling my digital records and documents, following crashes of two of my external hard disks, I have come across many documents that I hadn’t viewed in a long time. Among them are the papers I wrote for different classes as an undergraduate student at Grinnell College in the United States of America between 1990 and 1994. I have decided to reproduce some of them here in my personal blog.
I wrote this project report for my Statistics class. It appears that I collaborated with a classmate.
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Are foot length and lower arm length equal?
It has been said that the length of the lower arm of a person and the person’s foot are equal. This belief dates back to my childhood. In addition, it is commonly known that the sizes of women’s arm and foot are different from those of men. The purpose of this project was to determine how closely the two variables were related to each other and if the belief that the two are of equal lengths is valid. If the belief does not hold true, that is if there is a difference, how significant are the differences? We collected data from 58 females and 60 males and ran separate correlation, regression and significance tests on the male data, female data and the two data put together.
The data was collected not obtained from a simple random sample. We explained to people who we came across what we wanted and on receiving their written consent collected the data. Dorje collected most of his (15 men and 15 women) data from people living in his Hall. Kent collected his (15 men and 12 women) data from a few random people he came across, but mostly took the measurements from people whom he knew well. (See table 1 in appendix for the complete data.) Because body proportions are universal, we felt a convenience sample was sufficient for our purposes. The measurements were made using a measuring tape in Dorje’s case and a ruler in Kent’s case. Each measurement was made three times to ensure accuracy. The averages of the three measurements were used in the tests.
Because we were predicting arm length equal to the foot length, regression tests without a constant were conducted. Results were mixed. The equality of the two variables would have been indicated by a slope of 1.000. However, only the women’s data showed a equality relationship: a 68%-confidence interval of the slope included 1.000. For the two data put together, the 95%-confidence interval include 1.000. As for the male data, even the 95%-confidence interval of the slope did not include 1.000 (see table 2). In all three cases, the correlations were very strong, as expected (see table 3). However, correlation of course does not tell us anything about the equality of the two variables. From these results, it seems the belief holds true only in the case of women. Statistical significance tests also show stronger and more conclusive results pointing in the same direction.
We ran tests of significance, T-tests, on the three categories of data; men, women and mixed. First, the differences in forearm and foot lengths were found. Then the Ttest was carried out assuming a null hypothesis that the difference is zero. Surprisingly different P-values resulted from the three tests.
The female group yielded a very high P-value of 0.91 (91%). This means that from the data collected there is a 91 percent chance that the difference between the two variables is due to chance variation from the predicted one. The null hypothesis is valid. In this study the null hypothesis states that difference in the variables are due to chance. The alternative hypothesis states that the difference is real, that is, the length of the forearm and the foot are not equal.
The data from the men give a much lower P-value of 0.042. This is just below R. A. Fisher’s famous 5% cutoff point for determining statistical significance. This indicates that the difference is real and not chance variation. The null hypothesis should be rejected in this case. The pooled data yielded a P-value of 0.15. In this case, the null hypothesis should be kept. However, it is questionable whether these two sets of data, (male and female), should be combined. Their P-values are very different, strongly indicating the relationship between college age males’ foot and forearm length, is different than that of college age females.
As for why are these relationships different, one can only speculate. It could be due to environment or genetics. Shoes for women are often designed for aesthetics and cramp the foot, while shoes for men are much more often designed for comfort. It is possible that the women’s feet are stunted to around the length of the forearm by poorly fitting shoes, whereas men’s feet grow to their natural length which is a little longer than the forearm. Maybe the process of natural selection favored men with longer feet because they could run faster.
This is a difficult hypothesis to verify, which brings up the question of practical importance. Knowing why the relational values are different does not bring about any great improvements for society. Neither does the knowledge of the relationships between feet and forearms for men and women. These data are not practically important, however they are interesting.
There is no causal relationship between forearms and feet. The length of a foot does not cause the length of the forearm, neither does the length of the forearm cause the length of the foot. They are related by a third super-ordinate causal factor: genetic coding. The lack of a causal relationship further decreases the practical purposes of the data. Doctors can not can not solve a forearm problem by manipulating the foot, or vice versa.
We divided the collection of data between the two of us. Both of us ran the various minitab statistical tests together. We discussed the results and than divided up the questions. Dorje answered the first four questions. Kent answered the last three. We put together the two parts and edited it. We also collaborated on the oral report.
