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 R2R^2 is a measure of how well the regression model explains the variation in the observed data. It is calculated using the formula:

R2=1SSESSTR^2 = 1 - \frac{SSE}{SST}

Where:

  • SSTSST is the total sum of squares (given as 203.20),
  • SSESSE is the sum of squares due to error (given as 41.62).

Let's plug the values into the formula:

R2=141.62203.20R^2 = 1 - \frac{41.62}{203.20}

R2=10.2048=0.7952R^2 = 1 - 0.2048 = 0.7952

Thus, the value of R2R^2 is 0.7952.

This means that 79.52% of the variation in the observed yy-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 yy-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:

  1. How is the total sum of squares SSTSST calculated in a regression analysis?
  2. What is the meaning of the sum of squares due to error SSESSE?
  3. How does the value of R2R^2 affect the interpretation of a regression model's effectiveness?
  4. Can R2R^2 ever be negative, and what would it indicate if so?
  5. How does adding more independent variables to a model generally affect R2R^2?

Tip: A higher R2R^2 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