Question

It may seem odd, but biologists can tell how old a lobster is by measuring the concentration of pigment in the lobster's eye. 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, which would be the case if there is a strong relationship between the concentration in the right eye and the concentration in the left eye. Pigment concentration (as a percentage of tissue sample) was measured in both eyes for 39 lobsters, resulting in the following summary quantities (based on data from a graph in the paper): $$ \begin{array}{cll} n=39 & \sum_{x}=88.8 & \sum y=86.1 \\ \sum x y=281.1 & \sum x^{2}=288.0 & \sum y^{2}=286.6 \end{array} $$ An alternative formula for calculating the correlation coefficient that doesn't involve calculating the z-scores is $$ r=\frac{\sum_{x y}-\frac{\left(\sum x\right)\left(\sum y\right)}{n}}{\sqrt{\sum x^{2}-\frac{\left(\sum x\right)^{2}}{n}} \sqrt{\sum y^{2}-\frac{\left(\sum y\right)^{2}}{n}}} $$ Use this formula to calculate the value of the correlation coefficient, and interpret this value.

Short answer

The correlation coefficient r is approximately -0.044 indicating a very weak negative linear relationship between pigment concentration in the right eye and the pigment concentration in the left eye.

Step-by-step solution

Substitution of given values into formula

For a start, insert the provided values into the correlation coefficient formula to obtain \( r=\frac{281.1-\frac{88.8 \cdot 86.1}{39}}{\sqrt{288.0-\frac{88.8^2}{39}}} \cdot \sqrt{286.6-\frac{86.1^2}{39}} \)

Calculation of each operation according to the formula

Now, calculate the expressions one by one. Begin with the numerator of the fraction, then move to the denominator. The result for the numerator calculation is approximately -0.976 and the denominator calculation is approximately 22.36. Consequently, \( r = \frac{-0.976}{22.36} \)

Final calculation and interpretation

Perform the final division and then approximate the final correlation coefficient r, yielding \( r = -0.044 \). This answer shows a very weak negative correlation.