###### Project 6.3 Optimum Feeding Rate

Starlings often feed in flocks, and their rate of feeding depends on the size of the flock. If the flock is too small, the birds are nervous and spend a lot of time watching for predators. If the flock is too large, the birds become overcrowded and fight each other, which interferes with feeding. Here are some data gathered at a feeding station. The data show the number of starlings in the flock and the total number of pecks per minute recorded at the station while the flock was feeding. (Source: Chapman & Reiss, 1992)

Number of starlings |
Pecks per minute |
Pecks per starling per minute |

\(1\) | \(9\) | |

\(2\) | \(26\) | |

\(3\) | \(48\) | |

\(4\) | \(80\) | |

\(5\) | \(120\) | |

\(6\) | \(156\) | |

\(7\) | \(175\) | |

\(8\) | \(152\) | |

\(9\) | \(117\) | |

\(10\) | \(180\) | |

\(12\) | \(132\) |

For each flock size, calculate the number of pecks per starling per minute. For purposes of efficient feeding, what flock size appears to be optimum? How many pecks per minute would each starling make in a flock of optimal size?

Plot the number of pecks per starling per minute against flock size. Do the data points appear to lie on (or near) a parabola?

The quadratic regression equation for the data is \(y = -0.45x^2 + 5.8x + 3.9\text{.}\) Graph this parabola on the same axes with the data points.

What are the optimum flock size and the maximum number of pecks per starling per minute predicted by the regression equation?