Consider the four points (x_1,y_1) = (-5,-10), (x_2,y_2) = (0,3), (x_3,y_3) = (2,11) and (x_4,y_4) = (3,14). The line that minimizes the empirical squared error can be expressed as y = \hat{a}+\hat{b}x, where
\hat{a} =
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\hat{b} =
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To find the line that minimizes the empirical squared error, we need to find the values of a hat and b hat. The formula for the line is y = a hat + b hat x.
To find a hat and b hat, we can use the method of least squares. This method minimizes the sum of the squared differences between the observed values (y_i) and the predicted values (a hat + b hat x_i).
Let's break down the steps to find a hat and b hat:
Step 1: Calculate the mean of x (x bar) and y (y bar).
x bar = (x_1 + x_2 + x_3 + x_4) / 4
y bar = (y_1 + y_2 + y_3 + y_4) / 4
Plugging in the given values:
x bar = (-5 + 0 + 2 + 3) / 4
y bar = (-10 + 3 + 11 + 14) / 4
Step 2: Calculate the differences between each x value and x bar (x_i - x bar).
(x_1 - x bar), (x_2 - x bar), (x_3 - x bar), (x_4 - x bar)
Plugging in the given values:
(x_1 - x bar) = -5 - x bar
(x_2 - x bar) = 0 - x bar
(x_3 - x bar) = 2 - x bar
(x_4 - x bar) = 3 - x bar
Step 3: Calculate the differences between each y value and y bar (y_i - y bar).
(y_1 - y bar), (y_2 - y bar), (y_3 - y bar), (y_4 - y bar)
Plugging in the given values:
(y_1 - y bar) = -10 - y bar
(y_2 - y bar) = 3 - y bar
(y_3 - y bar) = 11 - y bar
(y_4 - y bar) = 14 - y bar
Step 4: Calculate the sum of the product of the differences between x values and x bar and the differences between y values and y bar.
Σ((x_i - x bar)(y_i - y bar))
Plugging in the calculated differences:
Σ((x_i - x bar)(y_i - y bar)) = ((-5 - x bar)(-10 - y bar)) + ((0 - x bar)(3 - y bar)) + ((2 - x bar)(11 - y bar)) + ((3 - x bar)(14 - y bar))
Step 5: Calculate the sum of the squared differences between x values and x bar.
Σ((x_i - x bar)^2)
Plugging in the calculated differences:
Σ((x_i - x bar)^2) = (-5 - x bar)^2 + (0 - x bar)^2 + (2 - x bar)^2 + (3 - x bar)^2
Step 6: Solve for b hat using the formula:
b hat = Σ((x_i - x bar)(y_i - y bar)) / Σ((x_i - x bar)^2)
Step 7: Solve for a hat using the formula:
a hat = y bar - b hat * x bar
By completing these steps, we can find the values of a hat and b hat that minimize the empirical squared error.