New method of estimating rainfall. Refer to the Journal of Data Science (Apr. 2004) comparison of methods for estimating rainfall, first presented in Exercise 9.27 (p. 500). Consider the simple linear regression relating the rain gauge amount (y) to the artificial neural network rain estimate (x).
a. Test whether y is positively related to x. Use α = .10.
b. Construct a 90% confidence interval for β1. Interpret the result practically.
Exercise 9.27
New method of estimating rainfall. Accurate measurements of rainfall are critical for many hydrological and me teorological projects. Two standard methods of monitoring rainfall use rain gauges and weather radar. Both, however, can be contaminated by human and environmental interference. In the Journal of Data Science (Apr. 200-1 ). researchers employed artificial neural networks (i.e., computer-based mathematical models) to estimate rainfall at a meteorological station in Montreal. Rainfall estimates were made every 5 minutes over a 70-minute period by each of the three methods. The data (in millimeters) are listed in the next table.
RAINFALL
Time | Radar | Rain Gauge | Neural Network |
8:00 a.m. | 3.6 | 0 | 1.8 |
8:05 | 2.0 | 1.2 | 1.8 |
8:10 | 1.1 | 1.2 | 1.4 |
8:15 | 1.3 | 1.3 | 1.9 |
8:20 | 1.8 | 1.4 | 1.7 |
8:25 | 2.1 | 1.4 | 1.5 |
8:30 | 3.2 | 2.0 | 2.1 |
8:35 | 2.7 | 2.1 | 1.0 |
8:40 | 2.5 | 2.5 | 2.6 |
8:45 | 3.5 | 2.9 | 2.6 |
8:50 | 3.9 | 4.0 | 4.0 |
8:55 | 3.5 | 4.9 | 3.4 |
9:00 A.M. | 6.5 | 6.2 | 6.2 |
9:05 | 7.3 | 6.6 | 7.5 |
9:10 | 6.4 | 7.8 | 7.2 |
Source: Hessami, M. et al. “Selection of an artificial neural network model for the post-calibration of weather radar rainfall estimation,” Journal of Data Science, Vol. 2, No. 2, Apr, 2004. (Adapted from Figures 2 and 4.)
a. Propose a straight-line model relating rain gauge amount (y) to weather radar rain estimate (x).
b. Use the method of least squares to fit the model to the data in the RAINFALL file.
c. Graph the least squares line on a scattergram of the data. Is there visual evidence of a relationship between the two variables? Is the relationship positive or negative?
d. Interpret the estimates of the y-intercept and slope in the words of the problem.
e. Now consider a model relating rain gauge amount (y) to the artificial neural network rain estimate (x). Repeat parts a-d for this model.
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