Find the required linear model using​ least-squares regression.

Question

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.

Answer 1

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.