Correlation Coefficient

Until now we have
looked at the graphs of data points to determine if the data “looks like” a
linear function or a quadratic function. This visual inspection is called “**the
shape of the data**.” From there we also looked at the amount of error as an
indication of the predictive power of the model we choose: low error, good
power; high error, may be good in certain ranges.

The **correlation coefficient** is another method
available for deciding if the model is one of strong predictive power. This
coefficient measures the strength of the linear/quadratic relation between two
variables. The coefficient is called ** r**, and the graphing
calculator calculates the coefficient if you use the TI83/TI83+.

The correlation coefficient is between – 1 and 1. Generally
speaking, the closer to – 1 or 1 the coefficient is, the stronger the
relationship and the predictive power of the model.

Using the
Correlation Coefficient

The data in the table below represents the depth of water in a tub in increments of one minute.

Type the following data into the list section of the calculator (L1 for time, L2 for depth):

Time
(in minutes) |
Depth
(cm) |
Time
(in minutes) |
Depth
(cm) |

1 |
3 |
10 |
19 |

2 |
6 |
11 |
25 |

3 |
7 |
12 |
23 |

4 |
7 |
13 |
27 |

5 |
8 |
14 |
31 |

6 |
12 |
15 |
31 |

7 |
13 |
16 |
36 |

8 |
18 |
17 |
35 |

9 |
18 |
18 |
37 |

1. Get the scatter plot and look at the shape of the data. What type of model would you think to use with this data?

- Get
the line of best fit (linear regression) for the data. ___________________________

- What
is the
value for the linear regression? ______________*r-*

- What
interpretation can you make about this value and the model?

Utility companies
need to be able to predict the peak power load to be able to handle operations
efficiently. The peak power load is the maximum amount of power that must be
generated each day to meet demand. The data below give the daily high
temperature and peak power load for a 21-day period one summer in one city.

Temperature
(F deg.) |
Peak
Load (megawatts) |
Temperature
(F deg.) |
Peak
Load (megawatts) |

94 |
135.0 |
98 |
150.1 |

96 |
131.7 |
100 |
157.9 |

95 |
140.7 |
102 |
175.6 |

88 |
115.4 |
103 |
198.5 |

84 |
113.4 |
106 |
225.2 |

90 |
123.0 |
105 |
205.3 |

97 |
143.2 |
92 |
130.0 |

71 |
95.0 |
92 |
130.0 |

67 |
101.6 |
85 |
112.4 |

79 |
105.2 |
89 |
113.5 |

87 |
112.7 |
74 |
103.9 |

- Get
the line of best fit (linear regression) for the data. ___________________________

- Get
the quadratic regression for the data.
_________________________________

- What
conclusion can you make about the two correlation coefficients?

- What
model would have the strongest predictive power?

- Which
model should the utility company use to predict power demands in this
city?

- Predict
the peak power when the high temperature is 93 degrees F.

- The
plant’s maximum capacity is 250 megawatts. Predict the temperature at
which this peak power load is expected.