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.
(b) Assuming the trend continues, estimate the total sales for the year 2015.
Let's first find the linear model for the given data. We need to find the equation of the line in the form y = mx + b, where y is the total sales and x is the year.
Step 1: Calculate the averages of the x and y values.
x-bar = (2006 + 2007 + 2008 + 2009 + 2010 + 2011)/6 = 2008.5
y-bar = (177.8 + 180.1 + 181.7 + 182.9 + 183.9 + 184.7)/6 = 181.8
Step 2: Calculate the differences between each x value and the x-bar, and each y value and the y-bar.
x - x-bar: -2.5, -1.5, -0.5, 0.5, 1.5, 2.5
y - y-bar: -4, -1.7, -0.1, 1.1, 2.1, 2.9
Step 3: Calculate the squared differences for each x and y pair.
(-2.5)^2 = 6.25, (-1.5)^2 = 2.25, (-0.5)^2 = 0.25, (0.5)^2 = 0.25, (1.5)^2 = 2.25, (2.5)^2 = 6.25
(-4)^2 = 16, (-1.7)^2 = 2.89, (-0.1)^2 = 0.01, (1.1)^2 = 1.21, (2.1)^2 = 4.41, (2.9)^2 = 8.41
Step 4: Calculate the product of the differences for each pair (x - x-bar)(y - y-bar).
(-2.5)(-4) = 10, (-1.5)(-1.7) = 2.55, (-0.5)(-0.1) = 0.05, (0.5)(1.1) = 0.55, (1.5)(2.1) = 3.15, (2.5)(2.9) = 7.25
Step 5: Calculate the sums of the squared differences and the product of differences.
Σ(x - x-bar)^2 = 17.25
Σ(y - y-bar)^2 = 33.87
Σ(x - x-bar)(y - y-bar) = 23.55
Step 6: Calculate the slope of the line.
m = Σ(x - x-bar)(y - y-bar) / Σ(x - x-bar)^2 = 23.55 / 17.25 ≈ 1.365
Step 7: Calculate the y-intercept of the line.
b = y-bar - m * x-bar = 181.8 - 1.365 * 2008.5 ≈ -1557.8
The linear model for the given data is y = 1.365x - 1557.8.
(a) For x = 6 (corresponding to the year 2006), we can substitute this value into the linear model equation to find the total sales for that year.
y = 1.365 * 6 - 1557.8 ≈ -1554.6 billion dollars
The total sales for the year 2006 is approximately -1554.6 billion dollars. (Note: The negative sign here indicates a decrease in sales compared to the previous year.)
(b) To estimate the total sales for the year 2015, we substitute x = 2015 into the linear model equation.
y = 1.365 * 2015 - 1557.8 ≈ 1817.7 billion dollars
The estimated total sales for the year 2015 is approximately 1817.7 billion dollars.