Question

Give a brief answer, comment, or explanation for each of the following. a. What is the difference between \(e_{1}, e_{2}, \ldots, e_{n}\) and the \(n\) residuals? b. The simple linear regression model states that \(y=\alpha+\beta x\) c. Does it make sense to test hypotheses about \(b\) ? d. SSResid is always positive. e. A student reported that a data set consisting of \(n=6\) observations yielded residuals \(2,0,5,3,0\), and 1 from the least-squares line. f. A research report included the following summary quantities obtained from a simple linear regression analysis: $$ \sum(y-\bar{y})^{2}=615 \quad \sum(y-\hat{y})^{2}=731 $$

Short answer

a. Residuals are the difference between the observed and predicted values while \(e_{1}, e_{2}, ..., e_{n}\) represent these residuals. b. This is the formula for a simple linear regression, where \(y\) is the dependent variable, \(x\) the independent variable, \(\alpha\) the y-intercept, and \(\beta\) the slope. c. Yes, it does make sense to test hypotheses about \(b\). d. Yes, SSR or sum of squared residuals is always positive. e. The residuals indicate how much the data points deviate from the regression line. f. The provided quantities represent the total sum of squares and sum of squares of residuals and there seems to be an error as SSR should be less than TSS.

Step-by-step solution

Understanding Residuals

Residuals are the differences between the observed and predicted values. In a mathematical model, \(e_{1}, e_{2}, ..., e_{n}\) are the n residuals for each of the n observed data points. These residuals represent the unexplained variation in the data after using the model to predict the outcome.

Simple linear regression model

The simple linear regression model has an equation of the form \(y=\alpha+\beta x\). In this equation, \(y\) is the dependent variable we're trying to predict or explain, \(x\) is the independent variable we're using to make predictions, \(\alpha\) is the y-intercept, and \(\beta\) is the slope of the regression line, which shows the change in \(y\) corresponding to a 1-unit increase in \(x\). Thus, it tells us the relationship between the dependent and independent variables

Hypothesis testing

Hypothesis testing is a statistical procedure used to determine whether a certain hypothesis about a population parameter is true. In the context of regression analysis, it makes sense to test hypotheses about the regression coefficient, \(b\). We often would want to know whether the true regression coefficient is 0, indicating that there is no relationship between the predictor and the outcome.

SSResid

The sum of squared residuals (SSResid) measures the total squared deviation of the observed outcomes from their predicted values according to the regression line. It is always positive because squared numbers are always positive or zero.

Interpretation of Residuals

Residuals help assess the fit of the regression model. In this case, a student reported residuals \(2,0,5,3,0\) , and 1 from the least-squares line from an analysis of 6 observations. The variety in residual values indicates that the data points are spread around the regression line.

Summary of simple linear regression analysis

The research report includes two sum of squares - \(\sum(y-\bar{y})^{2}=615\) is the total sum of squares (TSS), which represents the total variation in the dependent variable, \(y\).\(\sum(y-\hat{y})^{2}=731\) is the sum of squares of residuals (SSR), this quantity normally should be less than TSS because it represent unexplained variation in the data after fitting the model. However, here it is greater than TSS which indicates some error in the calculations.