x = 1990, 1995, 2000, 2005, 2007
y = 11.8, 13.2, 14.6, 15.9, 17
To find the line of best fit for the given data, we can use the least-squares method. This involves finding the equation of a line that minimizes the sum of the squared differences between the actual y-values and the predicted y-values.
Step 1: Calculate the mean of x and y.
mean(x) = (1990 + 1995 + 2000 + 2005 + 2007) / 5 = 2001.4
mean(y) = (11.8 + 13.2 + 14.6 + 15.9 + 17) / 5 = 14.7
Step 2: Calculate the differences between each x-value and the mean of x, and between each y-value and the mean of y.
x_diff = [1990 - 2001.4, 1995 - 2001.4, 2000 - 2001.4, 2005 - 2001.4, 2007 - 2001.4]
= [-11.4, -6.4, -1.4, 3.6, 5.6]
y_diff = [11.8 - 14.7, 13.2 - 14.7, 14.6 - 14.7, 15.9 - 14.7, 17 - 14.7]
= [-2.9, -1.5, -0.1, 1.2, 2.3]
Step 3: Calculate the product of x_diff and y_diff.
product = [-11.4 * -2.9, -6.4 * -1.5, -1.4 * -0.1, 3.6 * 1.2, 5.6 * 2.3]
= [33.06, 9.6, 0.14, 4.32, 12.88]
Step 4: Calculate the squared differences of x_diff.
x_diff_squared = [(-11.4)^2, (-6.4)^2, (-1.4)^2, (3.6)^2, (5.6)^2]
= [129.96, 40.96, 1.96, 12.96, 31.36]
Step 5: Calculate the sum of x_diff_squared.
sum(x_diff_squared) = 129.96 + 40.96 + 1.96 + 12.96 + 31.36
= 217.2
Step 6: Calculate the slope of the line of best fit (b).
b = sum(product) / sum(x_diff_squared)
= (33.06 + 9.6 + 0.14 + 4.32 + 12.88) / 217.2
= 0.236
Step 7: Calculate the y-intercept of the line of best fit (a).
a = mean(y) - b * mean(x)
= 14.7 - 0.236 * 2001.4
= 14.7 - 472.55
= -457.85
Therefore, the equation of the line of best fit for the given data is:
y = 0.236x - 457.85