When asked o sate the simple linear regression model, a student wrote it as follows: \(E\{Y_i\} = \beta_0 + beta_1+X_i + \varepsilon_i\). Do you agree? Explain your reasoning.
A student declared: “Regression is a very powerful tool. We can isolate fixed and variable costs by fitting a linear regression model, even when we have no data for small lot sizes when working with housing prices”. Do you agree? Explain your reasoning.
For the linear model
\[Y_i = \beta_0 + \beta_1 X_i + \varepsilon_i\]
What is the implication for the regression function if \(\beta_1 = 0\) so that the model is \(Y_i = \beta_0 + \varepsilon_i\)? How would the regression function plot on a graph?
For the linear model
\[Y_i = \beta_0 + \beta_1 X_i + \varepsilon_i\]
It was found that a observation \(Y_i\) fell directly on the fitted regression line (i.e. \(Y_i = \hat Y_i\)). If this case was deleted, would the least squares regression line fitted to the remaining \(n-1\) cases be changed? [Hint: What is the contribution to the error for this observation?]
In a test of the alternatives \(H_0: \beta_1 \leq 0\) versus \(H_{\alpha}: \beta_1 > 0\), an analyst concluded \(H_0\). Does this conclusion imply that there is no linear association between \(X\) and \(Y\)? Explain.
Derive the formula for regression sum of squares
\[SSR = b_1^2 \sum \left(X_i - \bar X\right)^2\] given that we have
\[SSR = \sum \left(\hat Y_i - \bar Y\right)^2\]
A student fitted a linear regression for a class assignment. The student plotted the residuals \(e_i\) against \(Y_i\) and found a positive relation. When the residuals where plotted against the fitted values \(\hat Y_i\) the student found no relation. How could this arise? Which is the more meaningful plot?