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
180.6
182.9
184.5
185.7
186.7
187.5
(a) Find a linear model for the data with xequals
6
corresponding to the year 2006.
(b) Assuming the trend continues, estimate the total sales for the year 2017
.
Question content area bottom
Part 1
(a) The linear model for the data is yequals
enter your response herexplusenter your response here
.
(Round to two decimal places as needed.)
To find the linear model, we need to find the equation of the line that best fits the given data points. We can use the least-squares regression method to do this.
Let's label the year as x and the total sales as y.
We can represent the given data points as (x, y):
(2006, 180.6)
(2007, 182.9)
(2008, 184.5)
(2009, 185.7)
(2010, 186.7)
(2011, 187.5)
Now, we need to find the slope and y-intercept of the linear model.
First, let's calculate the average of x values and the average of y values:
x̄ = (2006 + 2007 + 2008 + 2009 + 2010 + 2011)/6 = 2008.5
ȳ = (180.6 + 182.9 + 184.5 + 185.7 + 186.7 + 187.5)/6 = 184.15
Next, let's calculate the deviations from the average for x and y:
Δx = x - x̄
Δy = y - ȳ
We get the following deviations:
Δx = [-2.5, -1.5, -0.5, 0.5, 1.5, 2.5]
Δy = [-3.55, -1.25, 0.35, 1.55, 2.55, 3.35]
The sum of the product of the deviations is:
Σ(Δx * Δy) = (-2.5 * -3.55) + (-1.5 * -1.25) + (-0.5 * 0.35) + (0.5 * 1.55) + (1.5 * 2.55) + (2.5 * 3.35) = 23.05
The sum of the squared deviations of x is:
Σ(Δx^2) = (-2.5)^2 + (-1.5)^2 + (-0.5)^2 + (0.5)^2 + (1.5)^2 + (2.5)^2 = 15
Using the least-squares regression formula, we can find the slope:
b = Σ(Δx * Δy) / Σ(Δx^2) = 23.05 / 15 ≈ 1.54
Now, substituting the slope and the averages into the equation of a line, we can find the y-intercept:
y = mx + b
184.15 = 1.54 * 2008.5 + b
184.15 = 3096.79 + b
b ≈ 184.15 - 3096.79
b ≈ -2912.64
So, the linear model for the data is y ≈ 1.54x - 2912.64.
For Part (a), when x = 6 (corresponding to the year 2006), we can substitute the value into the linear equation:
y ≈ 1.54(6) - 2912.64
y ≈ 9.24 - 2912.64
y ≈ -2903.40
Therefore, the linear model for the data with x equals 6 (year 2006) is y ≈ -2903.40.
For Part (b), assuming the trend continues, we can substitute the value x = 11 (corresponding to the year 2017) into the linear equation:
y ≈ 1.54(11) - 2912.64
y ≈ 16.94 - 2912.64
y ≈ -2895.70
Therefore, the estimated total sales for the year 2017 is approximately -2895.70 billion dollars.