In: Statistics and Probability
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
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)