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In: Statistics and Probability

It appears that over the past 50 years, the number of farms in the United States...

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 (y) by the number of farms (x). Discuss the slope and y-intercept of the model.

Year Number of Farms (millions) Average Size (acres)
1960 5.69 218
1965 4.70 260
1970 3.91 298
1975 3.32 342
1980 2.95 372
1985 2.52 419
1990 2.47 426
1995 2.31 446
2000 2.16 459
2005 2.07 472
2010 2.18 433
2015 2.10 443


(Do not round the intermediate values. Round your answers to 2 decimal places.)

y^= ______ +( _____ )x

Solutions

Expert Solution

We know that, Regression equation is given as:

Here, we need to find the coefficients , which are given as,

Year Average Size (acres) (Y) Number of Farms (millions) (X) (xi-x̅) (yi-y̅) (xi-x̅)2 (xi-x̅)*(yi-y̅)
1960 218 5.69 2.658333 -164.333 7.066736 -436.8526837
1965 260 4.7 1.668333 -122.333 2.783336 -204.0927181
1970 298 3.91 0.878333 -84.3333 0.771469 -74.07274569
1975 342 3.32 0.288333 -40.3333 0.083136 -11.62943349
1980 372 2.95 -0.08167 -10.3333 0.006669 0.843886511
1985 419 2.52 -0.51167 36.6667 0.261803 -18.76112939
1990 426 2.47 -0.56167 43.6667 0.315469 -24.52613129
1995 446 2.31 -0.72167 63.6667 0.520803 -45.94613729
2000 459 2.16 -0.87167 76.6667 0.759803 -66.82780939
2005 472 2.07 -0.96167 89.6667 0.924803 -86.22947949
2010 433 2.18 -0.85167 50.6667 0.725336 -43.15114119
2015 443 2.1 -0.93167 60.6667 0.868003 -56.52114419
Total: 4588 36.38 15.08737 -1067.766667
Mean: y̅=382.3333 x̅=3.0316667

Hence,

Therefore, Regression equation will be

Interpretation:

for intercept: If number of farms is 0, then the model predicts that the average size is approximately 596.89 acres

for slope: If number of farms increases by 1 million, then the model predicts that the average size will decrease by 70.77 acres.

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Here, I have fitted the line plot using Minitab (for reference)

orchestra answered 1 year ago
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