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
2006
2007
2008
2009
2020
2011

total sales
184.6
186.9
188.5
189.7
190.7
189.7
​(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 2017 .

Answer 1

To find the linear model for the data, we need to use the least-squares regression method. The formula for a linear model is y = mx + b, where y is the dependent variable (total sales), x is the independent variable (year), m is the slope of the line, and b is the y-intercept.

We will first calculate the slope, m, using the formula:
m = (nΣxy - ΣxΣy) / (nΣx^2 - (Σx)^2),

where n is the number of data points, Σxy is the sum of the products of each x and y value, Σx is the sum of the x values, Σy is the sum of the y values, and Σx^2 is the sum of the squares of the x values.

In this case, we have the following data:
x = [2006, 2007, 2008, 2009, 2020, 2011]
y = [184.6, 186.9, 188.5, 189.7, 190.7, 189.7]
n = 6

Calculating Σx:
Σx = 2006 + 2007 + 2008 + 2009 + 2010 + 2011 = 12051

Calculating Σy:
Σy = 184.6 + 186.9 + 188.5 + 189.7 + 190.7 + 189.7 = 1130.1

Calculating Σx^2:
Σx^2 = 2006^2 + 2007^2 + 2008^2 + 2009^2 + 2010^2 + 2011^2 = 24314836

Calculating Σxy:
Σxy = (2006 * 184.6) + (2007 * 186.9) + (2008 * 188.5) + (2009 * 189.7) + (2010 * 190.7) + (2011 * 189.7)
Σxy = 3687896 + 3726903 + 3770800 + 3814313 + 3847070 + 3770807
Σxy = 22698989

Calculating the slope, m:
m = (nΣxy - ΣxΣy) / (nΣx^2 - (Σx)^2) = (6 * 22698989 - 12051 * 1130.1) / (6 * 24314836 - (12051)^2)
m ≈ 1.422

Now, we calculate the y-intercept, b, using the formula:
b = (Σy - mΣx) / n

b = (1130.1 - 1.422 * 12051) / 6
b ≈ -2754.2

Therefore, the linear model for the data is y ≈ 1.422x - 2754.2.

(a) To find the total sales for the year 2006 (x = 6), we substitute x = 6 into the equation:
y ≈ 1.422(6) - 2754.2
y ≈ -2745.4 + 2754.2
y ≈ 8.8

The estimated total sales for the year 2006 is 8.8 billion dollars.

(b) To estimate the total sales for the year 2017 (x = 2017), we substitute x = 2017 into the equation:
y ≈ 1.422(2017) - 2754.2
y ≈ 2869.134 - 2754.2
y ≈ 114.934

The estimated total sales for the year 2017 is approximately 114.934 billion dollars.