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
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.)

Answer 1

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.