Which of the following statements accurately describe aspects of simple linear regression? (Select all that apply)
1. The least squares method minimizes the sum of squared errors (SSE) between the observed data points and the regression line.
2. In simple linear regression, the regression line always passes through all the data points in the scatter plot.
3. The error term (ε) in the population model accounts for all variables not included in the regression model that may influence the dependent variable.
4. The slope of the regression line is represented by the coefficient b1, and the intercept is represented by b0 in the regression equation.
Answer (2)
In simple linear regression, statements 1, 3, and 4 are accurate. Statement 1 refers to the least squares method minimizing the sum of squared errors, statement 3 addresses the error term accounting for unmeasured variables, and statement 4 correctly identifies the coefficients for slope and intercept. Statement 2 is incorrect as the regression line does not pass through all data points.
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In simple linear regression, we are trying to model the relationship between two variables by fitting a linear equation to observed data. One variable is considered to be the independent variable (often denoted as x ) and the other is the dependent variable (often denoted as y ).
Let's examine each of the options given:
The least squares method minimizes the sum of squared errors (SSE) between the observed data points and the regression line.
This statement is accurate. The least squares method is a standard approach in regression analysis that finds the line of best fit for a set of data points. It works by minimizing the sum of the squared differences (errors) between the observed values and the values predicted by the linear model. The goal is to make these differences as small as possible, which is why this method is widely used for linear regression.
In simple linear regression, the regression line always passes through all the data points in the scatter plot.
This statement is not accurate. The regression line represents the best fit for the entire dataset but does not necessarily pass through all the data points. It provides an approximation of the relationship between x and y , minimizing the overall error, but individual data points can still be off the line unless it perfectly describes the data (which is rare in real-world scenarios).
The error term ( ε ) in the population model accounts for all variables not included in the regression model that may influence the dependent variable.
This statement is accurate. In the population regression model, y = b 0 + b 1 x + ε , the error term ε represents the effect of all other variables that could affect y but are not included in the model. It embodies the randomness and variability not explained by the model.
The slope of the regression line is represented by the coefficient b 1 , and the intercept is represented by b 0 in the regression equation.
This statement is accurate. In the equation y = b 0 + b 1 x , b 0 represents the y-intercept, which is the predicted value of y when x = 0 , and b 1 represents the slope of the line, indicating the change in y for a one-unit change in x .
So, the correct statements are 1, 3, and 4.
