New method of estimating rainfall. Accurate measurements of rainfall are critical for many hydrological and meteorological 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. LO9 

2004), 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 table.

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 Based on 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.

c. Graph the least squares line on a scatterplot 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. Find and interpret the value of s for this regression.

f. Test whether y is linearly related to x. Use a = .01.

g. Construct a 99% confidence interval for b1. Interpret the result practically.

h. Now consider a model relating rain gauge amount

(y) to the artificial neural network rain estimate (x).

Repeat parts a–g for this model.