The table shows the data for 5 perch caught in a lake In Finland. For all calculations --- show...

Question:

The table shows the data for 5 perch caught in a lake In Finland. For all calculations --- show the formulas with values used.

Perch Weight (grams) Length (cm) Width (cm)
1 100 19.2 3.3
2 110 22.5 3.6
3 120 23.5 3.5
4 150 24 3.6
5 145 25.5 3.8

What is the {eq}r^2 {/eq} for Weight and Length?

Coefficient of Determination:

The coefficient of determination is the square of the correlation coefficient. The higher is the coefficient of determination the more accurately we can predict the dependent variable using the independent variable.

Answer and Explanation:

We have:

Perch Weight, x Length, y
1 100 19.2
2 110 22.5
3 120 23.5
4 150 24
5 145 25.5
Total 625 114.7


The calculation for the correlation coefficient:

Perch Weight, x Length, y xy x^2 y^2
1 100 19.2 1920 10000 368.64
2 110 22.5 2475 12100 506.25
3 120 23.5 2820 14400 552.25
4 150 24 3600 22500 576
5 145 25.5 3697.5 21025 650.25
Total 625 114.7 14512.5 80025 2653.39

{eq}\begin{align*} r & =\dfrac{(n\sum xy)-(\sum x\sum y)}{\sqrt{\left [(n\sum x^{2})-(\sum x)^{2} \right ]\left [(n\sum y^{2})-(\sum y)^{2} \right ]}} \\[2ex] & =\dfrac{(5)(14512.5)-(625)(114.7)}{\sqrt{\left [(5)(80025)-(625)^{2} \right ]\left [(5)(2653.39)-(114.7)^{2} \right ]}} \\[2ex] & \approx 0.8526 \end{align*} {/eq}


The coefficient of determination:

{eq}\begin{align*} r^2 & = r \\[2ex] & = (0.8526)^2 \\[2ex] & \approx \bf{0.7270} \end{align*} {/eq}


Learn more about this topic:

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Coefficient of Determination: Definition, Formula & Example

from

Chapter 8 / Lesson 11
18K

The coefficient of determination is an important quantity obtained from regression analysis. In this lesson, we will show how this quantity is derived from linear regression analysis, and subsequently demonstrate how to compute it in an example.


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