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 184.8|187.1 | 188.7|189.9|190.9| 191.7
(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 2017.
(a) The linear model for the data is y = [ ]x + [ ].
(Round to two decimal places as needed.)
To find the linear model using least-squares regression, we need to find the equation of the line that best fits the given data points.
First, let's label the year as x and the total sales as y. We can create a table with the given data:
Year (x) | Total Sales (y)
2006 | 184.8
2007 | 187.1
2008 | 188.7
2009 | 189.9
2010 | 190.9
2011 | 191.7
Now, we can calculate the mean of x and y:
mean of x = (2006 + 2007 + 2008 + 2009 + 2010 + 2011) ÷ 6 = 2008.5
mean of y = (184.8 + 187.1 + 188.7 + 189.9 + 190.9 + 191.7) ÷ 6 = 188.85
Next, we calculate the sum of the products of (x - mean of x) and (y - mean of y):
∑[(x - mean of x)(y - mean of y)] = (2006 - 2008.5)(184.8 - 188.85) + (2007 - 2008.5)(187.1 - 188.85) + (2008 - 2008.5)(188.7 - 188.85) + (2009 - 2008.5)(189.9 - 188.85) + (2010 - 2008.5)(190.9 - 188.85) + (2011 - 2008.5)(191.7 - 188.85)
= -5.7 + 3.79 + -0.52 + 2.79 + 4.17 + 5.27
= 9.66
Then, we calculate the sum of the squares of (x - mean of x):
∑[(x - mean of x)^2] = (2006 - 2008.5)^2 + (2007 - 2008.5)^2 + (2008 - 2008.5)^2 + (2009 - 2008.5)^2 + (2010 - 2008.5)^2 + (2011 - 2008.5)^2
= (-2.5)^2 + (-1.5)^2 + (-0.5)^2 + 0.5)^2 + (1.5)^2 + (2.5)^2
= 15
Using these values, we can calculate the slope of the line:
slope (b) = ∑[(x - mean of x)(y - mean of y)] / ∑[(x - mean of x)^2]
slope (b) = 9.66 / 15
slope (b) = 0.64
To find the y-intercept (a), we can use the formula:
a = mean of y - b * mean of x
a = 188.85 - 0.64 * 2008.5
a ≈ -36.88
So, the linear model for the data is y = 0.64x - 36.88.
For part (b), we can use the linear model to estimate the total sales for the year 2017.
x (year) = 2017
y (total sales) ≈ 0.64 * 2017 - 36.88
y ≈ 1290.88 - 36.88
y ≈ 1254
Therefore, the estimated total sales for the year 2017 is approximately 1254 billion dollars.
the answer was incorrect
I apologize for the mistake. Let's recalculate the linear model and the estimated total sales for the year 2017.
Using the same process as before, we find:
∑[(x - mean of x)(y - mean of y)] = 9.66
∑[(x - mean of x)^2] = 15
The slope (b) is:
b = ∑[(x - mean of x)(y - mean of y)] / ∑[(x - mean of x)^2]
b = 9.66 / 15
b = 0.644
The y-intercept (a) is:
a = mean of y - b * mean of x
a = 188.85 - 0.644 * 2008.5
a ≈ -1290.74
So, the linear model for the data is y = 0.644x - 1290.74.
To estimate the total sales for the year 2017 (x = 2017), we substitute x = 2017 into the linear model:
y ≈ 0.644(2017) - 1290.74
y ≈ 1299.37 - 1290.74
y ≈ 8.63
Therefore, the estimated total sales for the year 2017 is approximately 8.63 billion dollars.
To find the required linear model using least-squares regression, follow these steps:
Step 1: Calculate the values of x and y. In this case, the years 2006-2011 correspond to x-values 1-6, and the total sales in billions of dollars are the y-values.
x: 1, 2, 3, 4, 5, 6
y: 184.8, 187.1, 188.7, 189.9, 190.9, 191.7
Step 2: Calculate the mean values of x and y.
x̄ = (1 + 2 + 3 + 4 + 5 + 6) / 6 = 3.5
ȳ = (184.8 + 187.1 + 188.7 + 189.9 + 190.9 + 191.7) / 6 = 188.4
Step 3: Calculate the deviations from the mean for x and y.
x deviations (x - x̄):
-2.5, -1.5, -0.5, 0.5, 1.5, 2.5
y deviations (y - ȳ):
-3.6, -1.3, 0.3, 1.5, 2.5, 3.3
Step 4: Calculate the sum of the products of the deviations.
∑(x deviations * y deviations) = (-2.5)(-3.6) + (-1.5)(-1.3) + (-0.5)(0.3) + (0.5)(1.5) + (1.5)(2.5) + (2.5)(3.3) = 28.95
Step 5: Calculate the sum of the squared deviations for x.
∑(x deviations)^2 = (-2.5)^2 + (-1.5)^2 + (-0.5)^2 + (0.5)^2 + (1.5)^2 + (2.5)^2 = 17.5
Step 6: Calculate the slope of the linear model.
slope (b) = ∑(x deviations * y deviations) / ∑(x deviations)^2 = 28.95 / 17.5 ≈ 1.6557
Step 7: Calculate the y-intercept of the linear model.
y-intercept (a) = ȳ - slope * x̄ = 188.4 - 1.6557 * 3.5 ≈ 182.9679
So, the linear model for the given data is:
y = 1.66x + 182.97
(a) The linear model for the data is y = 1.66x + 182.97.
(b) To estimate the total sales for the year 2017, substitute x = 11 (since the year 2017 corresponds to x = 11) into the linear model:
y = 1.66 * 11 + 182.97
y ≈ 200.26
Therefore, the estimated total sales for the year 2017 is approximately $200.26 billion.