Chapter 5.1, Problem 14E

It may seem odd, but one of the ways biologists can tell how old a lobster is involves measuring the concentration of a pigment called neurolipofuscin in the eyestalk of a lobster. (We are not making this up!) The authors of the paper “Neurolipofuscin Is a Measure of Age in Panulirus argus, the Caribbean Spiny Lobster, in Florida” (Biological Bulletin [2007]: 55-66) wondered if it was sufficient to measure the pigment in just one eye stalk, which would be the case if there is a strong relationship between the concentration in the right and left eyestalks.

Pigment concentration (as a percentage of tissue sample) was measured in both eyes talks for 39 lobsters, resulting in the following summary quantities (based on data read from a graph that appeared in the paper):

$\begin{array}{ccc}n=39& \sum x=88.8& \sum y=86.1\end{array}$

$\begin{array}{ccc}\sum xy=281.1& \sum x^2=288.0& \sum y^2=286.6\end{array}$

An alternative formula for calculating the correlation coefficient that is based on raw data and is algebraically equivalent to the one given in the text is

$r=\frac{\sum xy-\frac{(\sum x)(\sum y)}{n}}{\sqrt{\sum x^2-\frac{{(\sum x)}^2}{n}}\sqrt{\sum y^2-\frac{{(\sum y)}^2}{n}}}$

Use this formula to calculate the value of the correlation coefficient, and interpret this value.