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
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 .
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.