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
totals sales
183.6
185.9
187.5
188.7
189.7
190.5
(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
(a) The linear model for the data is y= blank x+ blank
(Round to two decimal places as needed.)
First, we need to determine the slope and y-intercept of the linear model. We can use the formula for the slope of a line:
m = (nΣ(xy) - ΣxΣy) / (nΣ(x^2) - (Σx)^2)
where n is the number of data points, Σ is the sum, xy is the product of x and y, and x^2 is the square of x.
Using the given data, we can calculate the necessary values:
n = 6
Σx = 6 + 7 + 8 + 9 + 10 + 11 = 51
Σy = 183.6 + 185.9 + 187.5 + 188.7 + 189.7 + 190.5 = 1126.9
Σ(xy) = (6)(183.6) + (7)(185.9) + (8)(187.5) + (9)(188.7) + (10)(189.7) + (11)(190.5) = 10,632.10
Σ(x^2) = (6^2) + (7^2) + (8^2) + (9^2) + (10^2) + (11^2) = 591
Now we can substitute these values into the slope formula:
m = (6(10,632.10) - (51)(1126.9)) / (6(591) - (51)^2)
m = (-17472.60) / (-288) ≈ 60.75
So the slope of the linear model is approximately 60.75.
Next, we can substitute one of the data points into the equation of a line (y = mx + b) to solve for the y-intercept:
183.6 = (60.75)(6) + b
183.6 = 364.50 + b
b = 183.6 - 364.50
b ≈ -180.90
So the y-intercept of the linear model is approximately -180.90.
Now we can write the equation of the linear model:
y = 60.75x - 180.90
Substituting x = 6, we find the value for y in the year 2006:
y = 60.75(6) - 180.90
y ≈ 231.90
Therefore, the linear model for the data is y = 60.75x - 180.90, where x represents the year and y represents total sales in billions of dollars.
(b) To estimate the total sales for the year 2017, we can substitute x = 2017 into the linear model:
y = 60.75(2017) - 180.90
y ≈ 122,897.25
Therefore, the estimated total sales for the year 2017 is approximately 122,897.25 billion dollars.