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 Total Sales
2006 177.8
2007 180.1
2008 181.7
2009 182.9
2010 183.9
2011 184.7
(a) Find a linear model for the data with x=6 corresponding to the year 2006.
To find the linear model for the data, we need to find the equation of a line that best fits the given data using least squares regression.
First, we'll assign the year (x) as the independent variable and the total sales (y) as the dependent variable.
We can use the formula for the equation of a line: y = mx + b, where m is the slope of the line and b is the y-intercept.
To find the slope (m), we can use the formula:
m = (Σ(x*y) - (Σx * Σy) / (Σx^2 - n * (Σx^2)),
where Σ(x*y) is the sum of the products of x and y, Σx is the sum of x values, Σy is the sum of y values, and Σx^2 is the sum of squares of x values. n is the total number of data points, which in this case is 6.
Using the given data, we calculate the sums:
Σ(x*y) = (2006*177.8) + (2007*180.1) + (2008*181.7) + (2009*182.9) + (2010*183.9) + (2011*184.7) = 1203901.2
Σx = 2006 + 2007 + 2008 + 2009 + 2010 + 2011 = 12051
Σy = 177.8 + 180.1 + 181.7 + 182.9 + 183.9 + 184.7 = 1090.1
Σx^2 = 2006^2 + 2007^2 + 2008^2 + 2009^2 + 2010^2 + 2011^2 = 24135312
Plugging these values into the formula, we get:
m = (1203901.2 - (12051 * 1090.1)) / (24135312 - 6 * (12051^2))
= (1203901.2 - 13131076.51) / (24135312 - 87123852)
= -11917175.31 / -62988540
≈ 0.189476629
So, the slope (m) is approximately 0.1895.
To find the y-intercept (b), we can use the formula:
b = (Σy - m * Σx) / n
Using the values we calculated earlier:
b = (1090.1 - 0.1895 * 12051) / 6
= (1090.1 - 2285.9545) / 6
= -1195.8545 / 6
≈ -199.30909
So, the y-intercept (b) is approximately -199.3091.
The linear model for the data is:
y = 0.1895x - 199.3091.
To find the total sales (y) for the year 2006 (x=6), we substitute x=6 into the equation:
y = 0.1895(6) - 199.3091
= 1.137 - 199.3091
≈ -198.172
Therefore, the estimated total sales (in billions of dollars) for the aerospace industry in the year 2006 is approximately -198.172 billion dollars.