# 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}

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Chapter 8 / Lesson 11The 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.