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 180.5 182.8 184.4 185.6 186.6 187.4
(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.
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.