Probability & Statistics Honors |
Unit Test #4 |

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1. |
In the mid-1990s, data concerning the size (carets) and price (dollars) for small diamonds were used to establish the following model: $\widehat{\text{price}}=-259.63+3721.02\left(\text{size}\right)$ The residual for a particular diamond was $410.1586. If the size of that diamond was 0.18 carets, what was the actual price? |
(4) |

The predicted price for that diamond was $ $-259.63+3721.02\left(0.18\right)\approx 410.1536$; since $\text{residual}=\text{observed}-\text{predicted}$, $410.1586=\text{observed}-410.1536\Rightarrow \text{observed}=410.1586+410.1536=820.3122$ dollars.

2. |
Researchers at Brigham Young University collected many body measurements from participants in a study. A graph of some of the data is shown above. Identify and classify any unusual points in the plot. |
(3) |

The point near $\left(125,130\right)$ has leverage; the point near $\left(149,135\right)$ is influential.

3. |
An analyst obtained data from a random sample of household electric bills, and found the following relationship between the amount of the bill (dollars) and the average temperature (°F) for the month: $\widehat{\text{bill}}=185.22-1.5674\left(\text{temperature}\right)$ Use this model to predict the bill in a month where the average temperature was 65°F. |
(3) |

$185.22-1.5674\left(65\right)=83.339$ dollars.

A paper in the American Journal of Physical Anthropology provided some measurements of several skeletons—the length of the metacarpal bones in the palm (cm), and the height (cm). A portion of the collected data is given below.

Palm |
45 |
51 |
39 |
41 |
48 |
49 |
46 |
43 |
47 |

Height |
171 |
178 |
157 |
163 |
172 |
183 |
173 |
175 |
173 |

Use this information for all other questions on this test.

4. |
Construct a model useful for predicting the height of a skeleton from the length of the metacarpal bones. |
(4) |

$\widehat{\text{height}}=94.428+1.7\left(\text{palm}\right)$

5. |
Interpret the slope of your equation from [4]. |
(4) |

For each additional centimeter of metacarpal length in the palm, the model predicts an average increase of 1.7 cm in height.

6. |
Interpret the |
(4) |

When the length of the metacarpals in the palm are zero cm, the model predicts an average height of 94.428 cm.

7. |
Calculate and interpret the coefficient of determination for these data. |
(4) |

${r}^{2}=0.7327$; about 73.27% of the variation in height can be explained by a linear relationship with metacarpal length of bones in the palm.

8. |
Predict the height of a skeleton with 44 cm metacarpal bones. |
(3) |

$94.428+1.7\left(44\right)\approx 169.2116$ cm

9. |
Would it be appropriate to use this model to predict the metacarpal length for a skeleton that is 185 cm tall? |
(3) |

No—this model can only be used to predict height.

10. |
Construct a residual plot for these data. |
(5) |

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