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Power Regression Model Example

Side note:  Power regressions will not allow an independent variable value of zero.
This example does not utilize an independent variable value of zero.


antelope
The true antelopes are found only in Africa and Asia.  They range in size from 12" (30 cm. at the shoulder) pygmy antelopes to giant elands, which are over 6 feet tall (180 cm) at the shoulder.  Most antelopes are between 3 to 4 feet tall (90-120 cm) at the shoulder. The horns of antelopes, unlike the antlers of deer, are un-branched, are made of a shell with a bony core, and are not shed.  The majority of antelopes reside in Africa.

Data:  The data below represents the length and mid-shaft diameters of the humerus bones of African Antelopes.

Diameter (mm) Length (mm)
17.6 159.9
26.0 206.9
31.9 236.8
38.9 269.9
45.8 300.6
51.2 323.6
58.1 351.7
64.7 377.6
66.7 384.1
80.8 437.2
82.9 444.7
Task: Express answers to the nearest tenth.
  a.)

Prepare a scatter plot of the data.

  b.) Determine a power regression model equation to represent this data.
  c.) Graph the new equation.
  d.) Decide whether the new equation is a "good fit" to represent this data.
  e.) Extrapolate data:  What length will correspond to a diameter of 84 mm?
  f.) Interpolate data:  What length will correspond to a diameter of 47 mm?
  g.) What mid-shaft diameter will correspond to a length of 305.7 mm?
    
             
Step 1.  Enter the data into the lists. 
For basic entry of data, see Basic Commands.

ant1 

Step 2.  Create a scatter plot of the data. 
     Go to STATPLOT (2nd Y=) and choose the first plot.  Turn the plot ON, set the icon to Scatter Plot (the first one), set Xlist to L1 and Ylist to L2 (assuming that is where you stored the data), and select a Mark of your choice.
   ant2       ant3

ant4

Step 3.  Choose the Power Regression Model.
     Press STAT, arrow right to CALC, and arrow down to A: PwrReg.  Hit ENTER.  When PwrReg appears on the home screen, type the parameters L1, L2, Y1.  The Y1 will put the equation into Y= for you. 
 (Y1 comes from VARS → YVARS, #Function, Y1)

   ant5    ant6    

ant7
The power regression equation is
amath1
(answer to part a)

Step 4.  Graph the Power Regression Equation from Y1.
     ZOOM #9 ZoomStat to see the graph.

ant8
(answer to part b)

Step 5.  Is this model a "good fit"?
     The correlation coefficient, r, is .9999937121 which indicates a very strong correlation since it is close to 1.
     The coefficient of determination, r
2, is .99999250766 which means that 99% of the total variation in y can be explained by the relationship between x and y.
     Yes, it is a very "good fit".
   
 (answer to part c)

antelope humerus

Step 6.  Extrapolate:  (beyond the data set)
    
 Go to TBLSET (above WINDOW) and set the TblStart to 84.  Notice how setting the increment (the deltaTbl = 0.1) displays certain values to two decimal places. The calculator is trying to tell you that this value has been "rounded" to the nearest hundredth and should not be now rounded to the nearest tenth.
Always check the "full" Y1 value, as seen at the bottom of the screen, before rounding.

(See an alternate method in Step 7.)

 antgraphtabletenth   333
           (answer to part d -- the length will be 448.3 mm)

Step 7.  Interpolate: 
                         (within the data set)

     From the graph screen, hit TRACE, arrow up to obtain the power equation, type 47, hit ENTER, and the answer will appear at the bottom of the screen.

  ant9
             (answer to part e --
        the length will be 305.7 mm)

Step 8.  What mid-shaft diameter will correspond to a length of 305.7 mm?
In reference to the equation, go to TABLE (above GRAPH) and arrow down until you find a Y1 value equal to (or close to) the desired value. The answer will appear in the X column. 47 mm.
4

 


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