To find the linear regression equation that models the data, we can use the method of least squares.
Step 1: Calculate the mean of the x-values (years) and the mean of the y-values (prices).
Mean of x-values (years):
(2002 + 2003 + 2004 + 2005 + 2007 + 2008 + 2009 + 2010 + 2011) / 9 = 2007
Mean of y-values (prices):
(0.70 + 1.04 + 1.15 + 1.38 + 1.70 + 2.55 + 1.29 + 2.22 + 2.56) / 9 ≈ 1.58
Step 2: Calculate the sum of the products of each x-value minus the mean of x-values and each y-value minus the mean of y-values.
Sum of (xi - mean of x) * (yi - mean of y):
(2002 - 2007)(0.70 - 1.58) + (2003 - 2007)(1.04 - 1.58) + (2004 - 2007)(1.15 - 1.58) + (2005 - 2007)(1.38 - 1.58) + (2007 - 2007)(1.70 - 1.58) + (2008 - 2007)(2.55 - 1.58) + (2009 - 2007)(1.29 - 1.58) + (2010 - 2007)(2.22 - 1.58) + (2011 - 2007)(2.56 - 1.58)
Simplifying the above equation, we get:
Sum of (xi - mean of x) * (yi - mean of y) = 3.04
Step 3: Calculate the sum of the squares of each x-value minus the mean of x-values.
Sum of (xi - mean of x)^2:
(2002 - 2007)^2 + (2003 - 2007)^2 + (2004 - 2007)^2 + (2005 - 2007)^2 + (2007 - 2007)^2 + (2008 - 2007)^2 + (2009 - 2007)^2 + (2010 - 2007)^2 + (2011 - 2007)^2
Simplifying the above equation, we get:
Sum of (xi - mean of x)^2 = 18
Step 4: Calculate the slope of the regression line.
Slope (b) = Sum of (xi - mean of x) * (yi - mean of y) / Sum of (xi - mean of x)^2
Slope (b) = 3.04 / 18 ≈ 0.169
Step 5: Calculate the y-intercept of the regression line.
y-intercept (a) = mean of y - slope * mean of x
y-intercept (a) ≈ 1.58 - 0.169 * 2007 ≈ -237.103
Step 6: Write the linear regression equation using the slope and y-intercept.
Regression equation: y = 0.169x - 237.103
Rounding to the nearest thousandth, the linear regression equation that models the data is:
y ≈ 0.169x - 237.103
2002
$0.70
2003
$1.04
2004
$1.15
2005
$1.38
2007
$1.70
2008
$2.55
2009
$1.29
2010
$2.22
2011
$2.56
Find the linear regression equation that models the data above. Round to the nearest thousandth if needed
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