Chapter 5, Problem 63CR

Data on salmon availability (x) and the percentage of eagles in the air (y) were given in the previous exercise.

1. a. Calculate the correlation coefficient for these data.
2. b. Because the scatterplot of the original data appeared curved, transforming both the x and y values by taking square roots was suggested. Calculate the correlation coefficient for the variables $\sqrt{x}$ and $\sqrt{y}.$ How does this value compare with that calculated in Part (a)? Does this indicate that the transformation was successful in straightening the plot?

   The paper “Aspects of Food Finding by Wintering Bald Eagles” (The Auk [1983]: 477–484) examined the relationship between the lime that eagles spend aerially searching for food (indicated by the percentage of eagles soaring) and relative food availability. The accompanying data were taken from a scatterplot that appeared in this paper. Use x to denote salmon availability and y to denote the percentage of eagles in the air.

1. a. Draw a scatterplot for this data set. Would you describe the pattern in the plot as linear or curved?
2. b. One possible transformation that might lead to a more nearly linear plot involves taking the square root of both the x and y values. Explain why this might be a reasonable transformation.
3. c. Construct a scatterplot using the variables $\sqrt{x}$ and $\sqrt{y}.$ Is this scatterplot more nearly linear than the scatterplot in Part (a)?
4. c. Suggest another transformation that might be used to straighten the original plot.