FRQ 2 An important advance in ecological research is the Theory of Island Biogeography, which demonstrates a relationship between the size of islands and the number of species of different amphibians that can be found in the islands. The scatterplot of number of species versus island area (in thousands of square miles) is nonlinear. The graphs below show the results of two different transformations of the data. Model I plots the square of the number of species against area. Model II plots the square root of the number of species against area. Linear regression analyses are provided below. Model I: (Number of Species)? vs. Area Model II: /Number of species vs. Area 8000 - 10 9 7000 6000 - 5000 – 4000 – 3000 2000 1000 0- 20 40 60 80 100 120 20 40 60 80 100 120 Area (thousands of square miles) Area (thousands of square miles) Predictor Coef SE Coef P Predictor Coef SE Coef T P Constant 440.43 29.35 15.01 0.0000 Constant 3.62 0.835 4.34 0.0074 Area 63.892 6.55 9.754 0.0002 Area 0.059 0.017 3.47 0.0178 S = 133.38 R-Sq = 89.68 R-Sq (adj) = 91.5% S = 1.806 R-Sq = 70.9% R-Sq (adj) = 71.5% (a) Based on the information provided, which model best describes the relationship between area and number of species? Explain. (b) Based on your answer to part (a), give the equation of the least-squares regression line using the transformed variable(s). Be sure to define any variables you use. (c) Use the model from part (b) to predict the number of amphibian species on the island of Martinique, which has an area of 0.436 thousands of square miles. (d) The island of Martinique actually had 23 amphibian species. Calculate and interpret the residual for Martinique. (Number of species)2 (səpads jo jəquinN) ds
FRQ 2 An important advance in ecological research is the Theory of Island Biogeography, which demonstrates a relationship between the size of islands and the number of species of different amphibians that can be found in the islands. The scatterplot of number of species versus island area (in thousands of square miles) is nonlinear. The graphs below show the results of two different transformations of the data. Model I plots the square of the number of species against area. Model II plots the square root of the number of species against area. Linear regression analyses are provided below. Model I: (Number of Species)? vs. Area Model II: /Number of species vs. Area 8000 - 10 9 7000 6000 - 5000 – 4000 – 3000 2000 1000 0- 20 40 60 80 100 120 20 40 60 80 100 120 Area (thousands of square miles) Area (thousands of square miles) Predictor Coef SE Coef P Predictor Coef SE Coef T P Constant 440.43 29.35 15.01 0.0000 Constant 3.62 0.835 4.34 0.0074 Area 63.892 6.55 9.754 0.0002 Area 0.059 0.017 3.47 0.0178 S = 133.38 R-Sq = 89.68 R-Sq (adj) = 91.5% S = 1.806 R-Sq = 70.9% R-Sq (adj) = 71.5% (a) Based on the information provided, which model best describes the relationship between area and number of species? Explain. (b) Based on your answer to part (a), give the equation of the least-squares regression line using the transformed variable(s). Be sure to define any variables you use. (c) Use the model from part (b) to predict the number of amphibian species on the island of Martinique, which has an area of 0.436 thousands of square miles. (d) The island of Martinique actually had 23 amphibian species. Calculate and interpret the residual for Martinique. (Number of species)2 (səpads jo jəquinN) ds
Chapter4: Linear Functions
Section: Chapter Questions
Problem 8PT: Does Table 1 represent a linear function? If so, finda linear equation that models the data.
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Transcribed Image Text:FRQ 2
An important advance in ecological research is the Theory of Island Biogeography, which demonstrates a relationship
between the size of islands and the number of species of different amphibians that can be found in the islands. The
scatterplot of number of species versus island area (in thousands of square miles) is nonlinear. The graphs below show
the results of two different transformations of the data. Model I plots the square of the number of species against area.
Model II plots the square root of the number of species against area. Linear regression analyses are provided below.
Model I: (Number of Species)? vs. Area
Model II:
Number of species vs. Area
8000
7000-
6000 -
5000 -
4000 -
3000 -
2000 -
1000 -
20
40
60
80
100
120
20
40
60
80
100
120
Area (thousands of square miles)
Area (thousands of square miles)
Predictor
Сoef
SE Coef
T
P
Predictor
Сoef
SE Coef
T
P
Constant
440.43
29.35
15.01
0.0000
Constant
3.62
0.835
4.34
0.0074
Area
63.892
6.55
9.754
0.0002
Area
0.059
0.017
3.47
0.0178
S = 133.38
R-Sg = 89.6%
R-Sq (adj) = 91.5%
S = 1.806
R-Sg = 70.9%
R-Sq (adj) = 71.5%
(a) Based on the information provided, which model best describes the relationship between area and number of
species? Explain.
(b) Based on your answer to part (a), give the equation of the least-squares regression line using the transformed
variable(s). Be sure to define
any
variables
you use.
(c) Use the model from part (b) to predict the number of amphibian species on the island of Martinique, which has an
area of 0.436 thousands of square miles.
(d) The island of Martinique actually had 23 amphibian species. Calculate and interpret the residual for Martinique.
(Number of species)²
SQRT(Number of species)
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