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

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.)

Answer 1

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.