In this problem use the power rule to determine the pulse (beats per min) as a function of the weight (kg) of the animal. You are given the following data concerning six animals [1]:

Animal

Weight (kg)

Pulse

Mouse

0.017

450

Hamster

0.103

347

Guinea Pig

0.437

269

Goat

33

81

Man

68

65

Cattle

500

49

 

a. Let P be the pulse and w be the weight, then the power law expression relating the pulse to the weight is given by

P = kwa.

Use the power law under Excel's trendline to best fit the data above. Plot the data and the best power law fit and have Excel write the formula on your graph. (Note that you will be adding the model from Part d to this graph, so wait to enter only one graph in your Lab report.) Determine the sum of squares error between the data and the model. How well does the graph match the data? Find the percent error between the pulse given by the model and the actual data for each of the animals in the table above. (Assume that the weight in the table is accurate.) Which animal has the highest percent error and explain why you might expect this? Also, which animal has the lowest percent error and explain why this might be the case?

b. The concept we use to fit power law to data is fitting a straight line to the logarithms of the data. Thus, from an allometric model of the form y = kxa , we obtain the formula ln(y) = ln(k) + a ln(x). With the best fitting model from Part a, take the logarithm of the pulse and the logarithm of the weight. Use Excel's scatter plot and linear fit under trendline to see how this fits these data. Plot a graph of the logarithm of the data and the best straight line fit to these data. Show the formula for this straight line on your graph. Compare the coefficients obtained in this manner to the ones found in Part b. How well does the graph match the data?

c. Have Excel plot a log-log plot of the data and the trendline that you found in Part a. (This is done by editing the graph and selecting logarithmic scales for both the x and y axes, which is easily done by double clicking on the axes.) Do the data roughly fall on a straight line in this plot?

d. Use the applet below to find the nonlinear least squares best fit to the data. (There is a hyperlink in the allometric section discussing this nonlinear fit.) Minimize the sum of squares error, then write in your lab report the equation of the best model and the value for the sum of square errors. (Hint: The least sum of square errors is between 2475 and 2525.) Find the percent error between the pulse given by this model and the actual data for each of the animals in the table above. Simulate this model and add its graph to the graph produced in Parts a and c. Compare this model to the one given by Excel above. Which is the better model? Explain why differences arise.

Alternate Link

 
e. Use both models to find the missing entries in the table below. Discuss which estimates are the best and which are the worst. Write a brief paragraph that describes how pulse and weight are related based on these models.

Animal

Weight (kg)

Pulse

Rat

352

Opossum

187

Swine

100

Elephant

2500

 


[1]P. L. Altman and D. M. Dittmer, (eds.), Biology Data Book , Federation of American Societies for Experimental Biology, 1964.