Probability & Statistics Honors |
Unit Test #4 |

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1. |
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: $\widehat{\text{bill}}=185.22-1.5674\left(\text{temperature}\right)$ The residual for a particular month was -$44.7909. If the average temperature that month was 32.5°F, what was the amount of the electric bill? |
(4) |

When the average temperature is 32.5, the predicted bill is 134.2795. The residual is $\text{observed}-\text{expected}$, so $-44.7909=\text{observed}-134.2795\Rightarrow \text{observed}=134.2795-44.7909=89.4886$ dollars.

2. |
Researchers in California 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 circled point near $\left(75,110\right)$ has leverage and appears to be an outlier (making it influential); the circled point near $\left(72,129\right)$ is an outlier.

3. |
In the mid-1990s, data concerning the size (carets) and price (dollars) for small diamonds were used to establish the following model: $\widehat{\text{price}}=-259.63+3721.02\left(\text{size}\right)$ Use this model to predict the price of a 0.3 caret diamond. |
(3) |

$\widehat{\text{price}}=-259.63+3721.02\left(0.3\right)\approx 856.676$ dollars.

Statistics from a sample of Major League Baseball players were collected during the 1992 season. A portion of the collected data is given below.

Runs |
74 |
32 |
87 |
16 |
102 |
6 |
0 |
39 |
4 |
65 |
58 |
41 |
21 |
46 |
88 |

Hits |
129 |
51 |
169 |
26 |
163 |
25 |
3 |
64 |
8 |
144 |
144 |
55 |
56 |
106 |
188 |

Use this information for all other questions on this test.

4. |
Construct a model useful for predicting the number of runs a player had from the number of hits. |
(4) |

The least squares regression model is $\widehat{\text{runs}}=0.9974+0.4989\left(\text{hits}\right)$.

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

For each additional hit a player has, the model predicts an average increase of 0.4989 runs.

6. |
Interpret the |
(4) |

When a player has zero hits, the model predicts an average of 0.9974 runs.

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

${r}^{2}=0.92003$; about 92% of the variation in the number of runs can be explained by a linear relationship with the number of hits.

8. |
Predict the number of runs for a player with 75 hits. |
(3) |

$\widehat{\text{runs}}=0.9974+0.4989\left(75\right)\approx 38.41507$

9. |
Would it be appropriate to use this model to predict the number of hits for a player with 110 runs? |
(3) |

No. This model can only be used to predict runs from the number of hits.

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

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