Compute the sum of the squared residuals of the least-squares line for the given data.
x -5 -3 4 1 -1 -2 0 2 3 -4
y -10 -8 9 1 -2 -6 -1 3 6 -8
A) 7.624 B) 1.036 C) 2.097 D) 0
The Correct Answer and Explanation is :
The correct answer is A) 7.624.
To find the sum of the squared residuals of the least-squares line for the given data, we start by fitting a linear model to the data points provided. The linear model, or least-squares line, minimizes the sum of the squared differences between the observed values (y-values) and the values predicted by the linear model.
The steps involved are as follows:
- Calculate the Coefficients: We compute the coefficients of the least-squares line using polynomial regression (in this case, a linear regression which is a first-degree polynomial). The coefficients define the slope and intercept of the line that best fits the data in a least-squares sense.
- Generate Predictions: Using the coefficients obtained, we predict the y-values based on the x-values. These predictions represent where each corresponding x-value would fall on the fitted line.
- Compute Squared Residuals: The residuals are the differences between the observed y-values and the predicted y-values. Squaring these residuals prevents negative values from offsetting positive values and emphasizes larger discrepancies.
- Sum of Squared Residuals: Finally, we sum all the squared residuals. This sum is a measure of how well the line fits the data; a smaller sum indicates a better fit.
In this specific case, when calculated, the sum of the squared residuals for the fitted line is approximately 7.624. This indicates the cumulative squared error between the fitted line and the actual data points.
Now, let’s generate an image depicting this scenario, showing the data points, the least-squares line, and the residuals visually.
The image above illustrates the scatter plot of your data points, the least-squares line fitted through them, and the vertical line segments representing the residuals (the distances between the data points and the line). This visual helps demonstrate how the least-squares line tries to minimize these residuals across all points.