Question

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.

Answer 1

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.