Compute the standard error of the estimate for the data below.
X values 3 –2 2 5 10
Y values 4 6 –2 0 –3
To compute the standard error of the estimate, we need to perform linear regression on the given data and find the equation of the regression line.
Step 1: Calculate the mean of the X values and the mean of the Y values.
Mean of X (X̄) = (3 - 2 + 2 + 5 + 10) / 5 = 3.6
Mean of Y (Ȳ) = (4 + 6 - 2 + 0 - 3) / 5 = 1
Step 2: Calculate the sum of the products of the differences between each X value and the mean of X (Xi - X̄) and the corresponding differences between each Y value and the mean of Y (Yi - Ȳ).
Σ((Xi - X̄) * (Yi - Ȳ)) = (3 - 3.6) * (4 - 1) + (-2 - 3.6) * (6 - 1) + (2 - 3.6) * (-2 - 1) + (5 - 3.6) * (0 - 1) + (10 - 3.6) * (-3 - 1)
Σ((Xi - X̄) * (Yi - Ȳ)) = -0.6 * 3 + (-6.6) * 5 + (-1.6) * (-3) + 1.4 * (-1) + 6.4 * (-4)
Σ((Xi - X̄) * (Yi - Ȳ)) = -1.8 - 33 - 4.8 - 1.4 - 25.6
Σ((Xi - X̄) * (Yi - Ȳ)) = -66.6
Step 3: Calculate the sum of the squares of the differences between each X value and the mean of X (Xi - X̄).
Σ((Xi - X̄)^2) = (3 - 3.6)^2 + (-2 - 3.6)^2 + (2 - 3.6)^2 + (5 - 3.6)^2 + (10 - 3.6)^2
Σ((Xi - X̄)^2) = (-0.6)^2 + (-5.6)^2 + (-1.6)^2 + (1.4)^2 + (6.4)^2
Σ((Xi - X̄)^2) = 0.36 + 31.36 + 2.56 + 1.96 + 40.96
Σ((Xi - X̄)^2) = 77.2
Step 4: Calculate the slope of the regression line (b), which is equal to Σ((Xi - X̄) * (Yi - Ȳ)) divided by Σ((Xi - X̄)^2).
b = Σ((Xi - X̄) * (Yi - Ȳ)) / Σ((Xi - X̄)^2)
b = -66.6 / 77.2
b = -0.861
Step 5: Calculate the intercept of the regression line (a), which is equal to the mean of Y minus the slope times the mean of X.
a = Ȳ - b * X̄
a = 1 - (-0.861) * 3.6
a = 4.102
The equation of the regression line is Y = 4.102 - 0.861X.
Now that we have the equation of the regression line, we can compute the standard error of the estimate.
Step 6: Calculate the predicted Y values (Y') for each X value using the equation of the regression line.
Y' = 4.102 - 0.861 * X
X = 3: Y' = 4.102 - 0.861 * 3 = 1.619
X = -2: Y' = 4.102 - 0.861 * -2 = 6.824
X = 2: Y' = 4.102 - 0.861 * 2 = 2.38
X = 5: Y' = 4.102 - 0.861 * 5 = -0.273
X = 10: Y' = 4.102 - 0.861 * 10 = -4.748
Step 7: Calculate the sum of the squared differences between each predicted Y value (Y') and the corresponding actual Y value (Y).
Σ(Yi - Y')^2 = (4 - 1.619)^2 + (6 - 6.824)^2 + (-2 - 2.38)^2 + (0 - (-0.273))^2 + (-3 - (-4.748))^2
Σ(Yi - Y')^2 = 3.361 + 0.676 + 19.776 + 0.072 + 2.625
Σ(Yi - Y')^2 = 26.51
Step 8: Calculate the standard error of the estimate (S).
S = √(Σ(Yi - Y')^2 / (n - 2))
S = √(26.51 / (5 - 2))
S = √(26.51 / 3)
S = √8.837
S = 2.974
Therefore, the standard error of the estimate for the given data is approximately 2.974.