Math Problem Statement
A scientific study on fishing gives the following data table.
Fishing Lines Fish Caught 4
13
5
15
7
25
11
29
12
26
Using technology, it was determined that the total sum of squares
(
S
S
T
)
was
203.20
and the sum of squares due to error
(
S
S
E
)
was
41.62
. Calculate
R
2
and determine its meaning. Round your answer to four decimal places.
Select the correct answer below:
0.7952
Therefore,
79.52
%
of the variation in the observed
y
-values can be explained by the estimated regression equation.
0.7952
Therefore,
0.7952
%
of the variation in the observed
y
-values can be explained by the estimated regression equation.
0.2048
Therefore,
20.48
%
of the variation in the observed
y
-values can be explained by the estimated regression equation.
0.2048
Therefore,
0.2048
%
of the variation in the observed
y
-values can be explained by the estimated regression
Solution
The coefficient of determination is a measure of how well the regression model explains the variation in the observed data. It is calculated using the formula:
Where:
- is the total sum of squares (given as 203.20),
- is the sum of squares due to error (given as 41.62).
Let's plug the values into the formula:
Thus, the value of is 0.7952.
This means that 79.52% of the variation in the observed -values (fish caught) can be explained by the estimated regression equation.
The correct answer is:
0.7952
Therefore, 79.52% of the variation in the observed -values can be explained by the estimated regression equation.
Would you like further details or have any questions?
Here are some relative questions to consider:
- How is the total sum of squares calculated in a regression analysis?
- What is the meaning of the sum of squares due to error ?
- How does the value of affect the interpretation of a regression model's effectiveness?
- Can ever be negative, and what would it indicate if so?
- How does adding more independent variables to a model generally affect ?
Tip: A higher indicates a better fit of the model to the data, but it doesn't necessarily mean the model is the best. Always check other statistical measures and the context.
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Math Problem Analysis
Mathematical Concepts
Statistics
Regression Analysis
Coefficient of Determination (R^2)
Formulas
R^2 = 1 - SSE/SST
Theorems
Theorem of Least Squares
Suitable Grade Level
Grades 11-12
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