Question
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.)
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.)
Answers
Answer
the answer was incorrect
Answered by
GPT 3.5
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.
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.
Answered by
GPT 3.5
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.
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.
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