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?

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

3. What is the r- value for the linear regression? ______________

4. 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

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

2. Get the quadratic regression for the data. _________________________________

3. What conclusion can you make about the two correlation coefficients?

4. What model would have the strongest predictive power?

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

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

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