## Related questions with answers

An observational study of 19 managers from a medium-sized manufacturing plant investigated which activities successful managers perform (Journal of Applied Behavioral Science, August 1985). To measure success, the researchers devised an index based on the manager's length of time in the organization and his or her level within the firm; the higher the index, the more successful the manager. The table on the following page presents data that can determine whether managerial success is related to the extensiveness of a manager's network-building interactions with people outside the manager's work unit. Such interactions include phone and face-to-face meetings with customers and suppliers, attending outside meetings, and doing public relations work.

Manager | Manager Success Index, $y$ | Number of Interactions with Outsiders, $x$ |
---|---|---|

1 | 40 | 12 |

2 | 73 | 71 |

3 | 95 | 70 |

4 | 60 | 81 |

5 | 81 | 43 |

6 | 27 | 50 |

7 | 53 | 42 |

8 | 66 | 18 |

9 | 25 | 35 |

10 | 63 | 82 |

11 | 70 | 20 |

12 | 47 | 81 |

13 | 80 | 40 |

14 | 51 | 33 |

15 | 32 | 45 |

16 | 50 | 10 |

17 | 52 | 65 |

18 | 30 | 20 |

19 | 42 | 21 |

**i**. In the context of this problem, decide the value of $x$ for which the associated prediction interval for $y$ is the narrowest.

Solution

VerifiedWe need to determine the $x$-value for which the prediction interval for $y$ is the most narrow.

The boundaries of the prediction interval for $y$ are $\hat{y}\pm E$ with $E=t_c s \sqrt{1+\dfrac{1}{n}+\dfrac{(x_0-\overline{x})^2}{S_{xx}}}$.

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