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: 178.8,181.1,182.7,183.9,184.9,185.7
(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 required linear model using least-squares regression, we will use the formula for a linear regression equation:
y = mx + b
where y is the total sales and x is the year.
First, we need to calculate the slope (m) and y-intercept (b) using the given data points.
Year: 2006, 2007, 2008, 2009, 2010, 2011
Total Sales: 178.8, 181.1, 182.7, 183.9, 184.9, 185.7
Step 1: Calculate the mean of the x-values (years) and the mean of the y-values (total sales).
Mean of x (years) = (2006 + 2007 + 2008 + 2009 + 2010 + 2011) / 6 = 2011 / 6 = 3351 / 6 = 558.5
Mean of y (total sales) = (178.8 + 181.1 + 182.7 + 183.9 + 184.9 + 185.7) / 6 = 1096.1 / 6 = 182.68
Step 2: Calculate the deviations from the mean for both x and y and multiply them.
(2006 - 558.5)(178.8 - 182.68) = -5523.8
(2007 - 558.5)(181.1 - 182.68) = -1360.58
(2008 - 558.5)(182.7 - 182.68) = 0.04
(2009 - 558.5)(183.9 - 182.68) = 1492.12
(2010 - 558.5)(184.9 - 182.68) = 754.24
(2011 - 558.5)(185.7 - 182.68) = 5543.34
Step 3: Calculate the deviations from the mean for x squared and sum them.
(2006 - 558.5)^2 = 8891952.25
(2007 - 558.5)^2 = 8809120.25
(2008 - 558.5)^2 = 8736544.25
(2009 - 558.5)^2 = 8674224.25
(2010 - 558.5)^2 = 8622160.25
(2011 - 558.5)^2 = 8580352.25
Step 4: Calculate the slope (m).
m = Σ[(x - mean of x)(y - mean of y)] / Σ[(x - mean of x)^2]
m = (-5523.8 + (-1360.58) + 0.04 + 1492.12 + 754.24 + 5543.34) / (8891952.25 + 8809120.25 + 8736544.25 + 8674224.25 + 8622160.25 + 8580352.25)
m = 1408.36 / 52925453.5
m ≈ 0.0000266
Step 5: Calculate the y-intercept (b).
b = mean of y - (m * mean of x)
b = 182.68 - (0.0000266 * 558.5 )
b ≈ 182.666
The linear model for the data is y = 0.0000266x + 182.666
(a) To find the linear model for the data with x = 6 corresponding to the year 2006, we substitute x = 6 into the equation.
y = 0.0000266(6) + 182.666
y ≈ 182.8 billion dollars
(b) To estimate the total sales for the year 2015, we substitute x = 2015 into the equation.
y = 0.0000266(2015) + 182.666
y ≈ 182.78 billion dollars
So, the estimated total sales for the year 2015 is approximately 182.78 billion dollars.
y = __x + __
The linear model equation, using least-squares regression for the given data, is:
y = 0.0000266x + 182.666
no
Apologies for the incorrect response.
To find the linear model using least-squares regression, we need to calculate the slope (m) and y-intercept (b) of the equation y = mx + b.
Step 1: Calculate the mean of the x-values (years) and the mean of the y-values (total sales).
Mean of x (years) = (2006 + 2007 + 2008 + 2009 + 2010 + 2011) / 6 = 12051 / 6 = 2008.5
Mean of y (total sales) = (178.8 + 181.1 + 182.7 + 183.9 + 184.9 + 185.7) / 6 = 1097.1 / 6 = 182.85
Step 2: Calculate the deviations from the mean for both x and y.
(2006 - 2008.5) = -2.5
(2007 - 2008.5) = -1.5
(2008 - 2008.5) = -0.5
(2009 - 2008.5) = 0.5
(2010 - 2008.5) = 1.5
(2011 - 2008.5) = 2.5
(178.8 - 182.85) = -4.05
(181.1 - 182.85) = -1.75
(182.7 - 182.85) = -0.15
(183.9 - 182.85) = 1.05
(184.9 - 182.85) = 2.05
(185.7 - 182.85) = 2.85
Step 3: Calculate the product of the deviations for x and y.
(-2.5)(-4.05) = 10.125
(-1.5)(-1.75) = 2.625
(-0.5)(-0.15) = 0.075
(0.5)(1.05) = 0.525
(1.5)(2.05) = 3.075
(2.5)(2.85) = 7.125
Step 4: Calculate the sum of the deviations and deviations squared for x.
Σ(x - mean of x) = -2.5 + (-1.5) + (-0.5) + 0.5 + 1.5 + 2.5 = 0
Σ(x - mean of x)^2 = (-2.5)^2 + (-1.5)^2 + (-0.5)^2 + 0.5^2 + 1.5^2 + 2.5^2 = 22.5
Step 5: Calculate the slope (m).
m = Σ((x - mean of x)(y - mean of y)) / Σ((x - mean of x)^2)
m = (10.125 + 2.625 + 0.075 + 0.525 + 3.075 + 7.125) / 22.5
m = 23.55 / 22.5
m ≈ 1.047
Step 6: Calculate the y-intercept (b).
b = mean of y - (m * mean of x)
b = 182.85 - (1.047 * 2008.5)
b ≈ -1003.763
The linear model for the data is:
y = 1.047x - 1003.763
(a) To find the linear model for the data with x = 6 corresponding to the year 2006, we substitute x = 6 into the equation.
y = 1.047(6) - 1003.763
y ≈ -997.884
(b) To estimate the total sales for the year 2015, we substitute x = 2015 into the equation.
y = 1.047(2015) - 1003.763
y ≈ 1082.732
So, the estimated total sales for the year 2015 is approximately 1082.732 billion dollars.
y = 1.35x + 171.41
I apologize for the previous incorrect response.
The correct linear model using least-squares regression for the given data is:
y = 1.35x + 171.41
(a) To find the linear model for the data with x = 6 corresponding to the year 2006, we substitute x = 6 into the equation:
y = 1.35(6) + 171.41
y = 8.10 + 171.41
y ≈ 179.51 billion dollars
(b) To estimate the total sales for the year 2015, we substitute x = 2015 into the equation:
y = 1.35(2015) + 171.41
y = 2715.25 + 171.41
y ≈ 2886.66 billion dollars
Therefore, the estimated total sales for the year 2015 is approximately 2886.66 billion dollars.