Find the required linear model using least-squares regression.
The table below gives the total sales (in billions of dollars) for the aerospace industry.
Year
2006
2007
2008
2009
2010
2011
Total Sales
180.6
182.9
184.5
185.7
186.7
187.5
(a) Find a linear model for the data with xequals
6
corresponding to the year 2006.
To find the linear model for the data, we need to use the least-squares regression method.
First, we need to assign the year as the independent variable (x) and the total sales as the dependent variable (y). We can create a table for the data:
Year (x) Total Sales (y)
2006 180.6
2007 182.9
2008 184.5
2009 185.7
2010 186.7
2011 187.5
Next, we need to calculate the values for the regression equation:
n = 6 (number of observations)
Σx = 6 + 7 + 8 + 9 + 10 + 11 = 51
Σy = 180.6 + 182.9 + 184.5 + 185.7 + 186.7 + 187.5 = 1107.9
Σxy = (6 * 180.6) + (7 * 182.9) + (8 * 184.5) + (9 * 185.7) + (10 * 186.7) + (11 * 187.5) = 6892.1
Σx^2 = (6^2) + (7^2) + (8^2) + (9^2) + (10^2) + (11^2) = 441
Now, we can calculate the slope (b) of the regression line:
b = [n * Σxy - Σx * Σy] / [n * Σx^2 - (Σx)^2]
b = [6 * 6892.1 - 51 * 1107.9] / [6 * 441 - (51)^2]
b = (41352.6 - 56557.29) / (2646 - 2601)
b = -151.69 / 45
b = -3.37
Finally, we can calculate the y-intercept (a) of the regression line:
a = (Σy - b * Σx) / n
a = (1107.9 - (-3.37) * 51) / 6
a = (1107.9 + 171.87) / 6
a = 1279.77 / 6
a = 213.29
Thus, the linear model for the data is:
y = 213.29 - 3.37x
Note: Since x = 6 corresponds to the year 2006, the linear model is in terms of the number of years after 2006.