Find the required linear model using​ least-squares regression. The table below gives the total sales​ (in billions of​ dollars) for the aerospace industry.

Question

Year 2006 2007 2008 2009 2010 2011
Total Sales 180.5 182.8 184.4 185.6 186.6 187.4
​(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 2015.

Answer 1

To find the linear model using least-squares regression, we need to find the equation of a line that best fits the given data points.

Let's assign the variables:
x = years since 2006 (x = 0 corresponds to the year 2006)
y = total sales in billions of dollars

Using the given data points:
(x,y) = (0, 180.5), (1, 182.8), (2, 184.4), (3, 185.6), (4, 186.6), (5, 187.4)

1. Find the slope (m) of the line:
To find the slope, we need to use the formula:
m = Σ((x - x̄)(y - ȳ))/ Σ(x - x̄)²
Where Σ indicates the sum of the values, x̄ is the mean of x values, and ȳ is the mean of y values.

Calculating the sums:
Σx = 0 + 1 + 2 + 3 + 4 + 5 = 15
Σy = 180.5 + 182.8 + 184.4 + 185.6 + 186.6 + 187.4 = 1107.3
Σ(x - x̄)(y - ȳ) = (0 - 2.5)(180.5 - 184.55) + (1 - 2.5)(182.8 - 184.55) + (2 - 2.5)(184.4 - 184.55) + (3 - 2.5)(185.6 - 184.55) + (4 - 2.5)(186.6 - 184.55) + (5 - 2.5)(187.4 - 184.55) = -47.825
Σ(x - x̄)² = (0 - 2.5)² + (1 - 2.5)² + (2 - 2.5)² + (3 - 2.5)² + (4 - 2.5)² + (5 - 2.5)² = 7.5

Substituting the values into the slope formula:
m = (-47.825) / 7.5 = -6.38

2. Find the y-intercept (b) of the line:
To find the y-intercept, we can use the formula:
b = ȳ - m * x̄
Substituting the values:
ȳ = (1107.3) / 6 = 184.55
x̄ = (15) / 6 = 2.5

b = 184.55 - (-6.38) * 2.5 = 201.37

3. Write the equation of the linear model:
The equation of the linear model is y = mx + b, where m is the slope and b is the y-intercept.
Substituting the values:
y = -6.38x + 201.37

(a) For x = 6 (corresponding to the year 2012), we can find the total sales by substituting the value into the equation:
y = -6.38(6) + 201.37
y = -38.28 + 201.37
y ≈ 163.09 billion dollars

(b) To estimate the total sales for the year 2015, we need to find the x-value that corresponds to that year. Since 2015 is 9 years after 2006, x = 9.
Substituting x = 9 into the equation:
y = -6.38(9) + 201.37
y = -57.42 + 201.37
y ≈ 143.95 billion dollars

Therefore, the estimated total sales for the year 2015 is approximately 143.95 billion dollars.