Suppose that we wish to fit a regression model for which the true regression line passes through the point (0, 0). The appropriate model is Y = βx + ε. Assume that we have n pairs of data (x₁, y₁), (x₂, y₂), . . . , (x_n, y_n).

(a) Find the least squares estimate of β.

(b) Fit the model Y = βx + e to the chloride concentrationroadway area data in Exercise 11-12. Plot the fitted model on a scatter diagram of the data and comment on the appropriateness of the model.

Exercised 11-12

An article in the Journal of Environmental Engineering (1989, Vol. 115(3), pp. 608€“619) reported the results of a study on the occurrence of sodium and chloride in surface streams in central Rhode Island. The following data are chloride concentration y (in milligrams per liter) and roadway area in the watershed x (in percentage).

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4.4 6.6 9.7 10.6 10.8 10.9 y 0.19 0.15 0.57 0.70 0.67 0.63 х 11.8 12.1 14.3 14.7 15.0 17.3 0.78 0.47 0.70 0.60 0.78 0.81 х 19.2 23.1 27.4 27.7 31.8 39.5 0.69 1.30 1.05 1.06 1.74 1.62 х