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
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