Question

Data on \(y=\) time to complete a task (in minutes) and \(x=\) number of hours of sleep on previous night were used to find the least squares regression line. The equation of the line was \(\hat{y}=12-0.36 x .\) For this data set, would the sum of squared deviations from the line \(y=12.5-0.5 x\) be larger or smaller than the sum of squared deviations from the least squares regression line? Explain your choice. (Hint: Think about the definition of the least- squares regression line.)

Short answer

The sum of squared deviations from the line \(y=12.5-0.5x\) would be larger than the sum of squared deviations from the least squares regression line \(y=12-0.36x\). This is based on the definition of the least squares regression line, which minimizes the sum of the squared deviations.

Step-by-step solution

Understand the least squares line

The least squares regression line minimizes the sum of the squared deviations, or errors, between the observed (real) and predicted values. In other words, it's the line that best fits a given set of data.

Analyze the given regression lines

You are provided with two regression lines: (1) the least squares regression line \(\hat{y}=12-0.36x\) and (2) another line \(y=12.5-0.5x\). You are asked to determine whether the sum of squared deviations from line 2 is larger or smaller than that from the least squares regression line.

Make a conclusion based on the property of least squares line

From the definition of the least squares regression line, it is known that this line minimizes the sum of squared deviations. Therefore, any other line would have a larger sum of squared deviations from the observed values than the least squares regression line. Therefore, the sum of squared deviations from the line \(y=12.5-0.5x\) would be larger than the sum of squared deviations from the least squares regression line \(y=12-0.36x\).

Key concepts

Sum of Squared Deviations

Understanding the sum of squared deviations is crucial when dealing with linear regression. This term refers to the sum of the squared differences between the observed data points and the values predicted by a regression line. In more practical terms, suppose you have a scatter plot showing the relationship between two variables. Picture a line through the scatter plot that attempts to show the trend in the data. The deviations are the vertical distances (also known as errors or residuals) between the data points and this line.

Why squaring the deviations? It serves two purposes: first, to ensure that negative and positive deviations do not cancel each other out; and second, to give more weight to larger deviations. This is because larger deviations suggest that the prediction is far off from the actual data points and should, therefore, bear a greater penalty.

For a given dataset, the least squares regression line is the one that has the smallest sum of squared deviations. It means this line is the best fit as it is closest to all the points overall, representing the trend most accurately. Any other line will naturally result in a larger sum of these squared deviations, hence validating the exercise's hypothesis.

Linear Regression

Linear regression is a foundational statistical method used to model and analyze the relationship between two quantitative variables. When data analysts perform linear regression, they are often looking for the equation of the least squares regression line. This equation has the form \(\hat{y} = a + bx\), where \(\hat{y}\) is the predicted value of the dependent variable, \(a\) represents the y-intercept, and \(b\) denotes the slope of the line. The slope offers insight into the relationship's direction and strength - a steeper slope indicates a stronger relationship between the independent and dependent variables.

Linear regression calculations revolve around finding the values for the slope and intercept that minimize the sum of squared deviations. This optimization is rooted in calculus and can be solved using statistical software or by applying related formulas. It is consistent in achieving its goal: fitting a line that sums up the trend in the data with the least amount of discrepancy. Thus, the exercise perfectly encapsulates linear regression as it highlights a comparison of the least squares regression line with another line to understand fit quality.

Prediction Errors

Prediction errors, also known as residuals, play a pivotal role in determining the accuracy of a linear regression model. These errors are the differences between the observed values of the dependent variable and the values predicted by the regression model. It’s the gap between reality and prediction. If we put this into context with the provided exercise, each point in the dataset has its own prediction error once we map it against the regression line.

Minimizing these errors across all data points is the key aim of a least squares regression line. The sum of these prediction errors squared comprises a critical metric for judging the efficiency of a regression line. In fact, the very method of 'least squares' is named after the objective of making the sum of the squares of these errors as small as possible.

It is important to understand that while prediction errors can never be completely eradicated, the process of linear regression ensures that these errors are reduced to the lowest possible values. As hinted in the exercise, a non-least squares regression line such as \(y=12.5-0.5x\) would indeed birth a heftier sum of squared prediction errors compared to the least squares line, underscoring its inferiority in terms of data prediction accuracy.