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
2010
2011
Total Sales
184.6
186.9
188.5
189.7
190.7
191.5
(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 using least-squares regression, we need to find the equation of the line that best fits the given data.
Let's label the independent variable (x) as the number of years since 2006, so x = 0 corresponds to the year 2006.
Next, we can assign x-values to each year:
2006: x = 0
2007: x = 1
2008: x = 2
2009: x = 3
2010: x = 4
2011: x = 5
The given table already provides the y-values (total sales):
Total Sales (y): 184.6, 186.9, 188.5, 189.7, 190.7, 191.5
To find the linear model, we can use the equation of a line: y = mx + b, where m is the slope and b is the y-intercept.
To find the slope (m) of the line, we can use the formula:
m = (n∑(xy) - ∑x∑y) / (n∑x^2 - (∑x)^2)
where n is the number of data points, ∑ (sum) represents the sum of the indicated values, x is the independent variable, and y is the dependent variable.
Let's calculate the values we need for the slope calculation:
n = 6 (number of data points)
∑(xy) = (0*184.6) + (1*186.9) + (2*188.5) + (3*189.7) + (4*190.7) + (5*191.5) = 1845.5
∑x = 0 + 1 + 2 + 3 + 4 + 5 = 15
∑y = 184.6 + 186.9 + 188.5 + 189.7 + 190.7 + 191.5 = 1131.9
∑x^2 = (0^2) + (1^2) + (2^2) + (3^2) + (4^2) + (5^2) = 55
Now we can calculate the slope (m):
m = (6*1845.5 - 15*1131.9) / (6*55 - 15^2)
m = (11073 - 16978.5) / (330 - 225)
m = -589.5 / 105
m ≈ -5.61
Now that we have the slope, we can find the y-intercept (b) by using the formula:
b = (∑y - m∑x) / n
Using the values we have:
b = (1131.9 - (-5.61) * 15) / 6
b = (1131.9 + 84.15) / 6
b ≈ 201.4
Therefore, the linear model for the data is given by the equation:
y = -5.61x + 201.4
To estimate the total sales for the year 2017 (x = 2017 - 2006 = 11), we can substitute this value into the equation:
y = -5.61 * 11 + 201.4
y ≈ 134.49
Therefore, the estimated total sales for the year 2017 would be approximately $134.49 billion.