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 Total Sales 183.1 185.4 187.0 188.2 189.2 190.0 (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 = 1.35x + 175.71 . (Round to two decimal places as needed.) (b) The estimated total sales for the year 2017 is $ billion. 2011 (Round to one decimal place as needed.)
(a) The linear model for the data can be found using the least-squares regression method. The equation for a linear model is of the form y = mx + b, where m is the slope and b is the y-intercept.
To find the slope, we use the formula:
m = (n∑(xy) - ∑x∑y) / (n∑(x^2) - (∑x)^2)
Calculating the sums:
∑x = 6 + 7 + 8 + 9 + 10 = 40
∑y = 183.1 + 185.4 + 187.0 + 188.2 + 189.2 + 190.0 = 1122.9
∑(xy) = (6*183.1) + (7*185.4) + (8*187.0) + (9*188.2) + (10*189.2) = 11155.1
∑(x^2) = (6^2) + (7^2) + (8^2) + (9^2) + (10^2) = 330
n is the number of data points, which is 5 in this case.
Substituting the values into the formula:
m = (5*11155.1 - 40*1122.9) / (5*330 - 40^2)
m = (55775.5 - 44916) / (1650 - 1600)
m = 10859.5 / 50
m = 217.19
The slope (m) is 217.19.
To find the y-intercept (b), we can use the formula:
b = (∑y - m∑x) / n
Substituting the values into the formula:
b = (1122.9 - 217.19*40) / 5
b = (1122.9 - 8687.6) / 5
b = -7564.7 / 5
b = -1512.94
The y-intercept (b) is -1512.94.
Therefore, the linear model for the data is y = 217.19x - 1512.94.
(b) To estimate the total sales for the year 2017, we substitute x = 11 into the linear model:
y = 217.19(11) - 1512.94
y = 2389.09 - 1512.94
y = 876.15
The estimated total sales for the year 2017 is $876.15 billion.