Question

It appears that over the past 50 years, the number of farms in the United States declined while the average size of farms increased. The following data provided by the U.S. Department of Agriculture show five-year interval data for U.S. farms. Use these data to develop the equation of a regression line to predict the average size of a farm by the number of farms Discuss the slope and y-intercept of the model.

Year	Number of Farms (millions)	Average Size (acres)
1960	5.70	210
1965	4.68	257
1970	4.00	302
1975	3.34	345
1980	2.98	378
1985	2.51	421
1990	2.47	426
1995	2.29	436
2000	2.15	460
2005	2.07	466
2010	2.16	431
2015	2.11	446

Answer 1

The independent variable is X, and the dependent variable is Y. In order to compute the regression coefficients, the following table needs to be used:

	X	Y	X*Y	X2	Y2
	5.7	210	1197	32.49	44100
	4.68	257	1202.76	21.9024	66049
	4	302	1208	16	91204
	3.34	345	1152.3	11.1556	119025
	2.98	378	1126.44	8.8804	142884
	2.51	421	1056.71	6.3001	177241
	2.47	426	1052.22	6.1009	181476
	2.29	436	998.44	5.2441	190096
	2.15	460	989	4.6225	211600
	2.07	466	964.62	4.2849	217156
	2.16	431	930.96	4.6656	185761
	2.11	446	941.06	4.4521	198916
Sum =	36.46	4578	12819.51	126.0986	1825508

Therefore, based on the above calculations, the regression coefficients (the slope m, and the y-intercept n) are obtained as follows:

Therefore, we find that the regression equation is:

Predict the average size of a farm by the number of farms. Here when x=0, y is 597.6562, the y-intercept which shows the minimum farm size. As the number of farms increases by 1, farm size decrease by a factor of 71.143 (which is the slope)

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