Assignment 7

Exercise 1 (7 points)

A student stated: “Adding predictor variables to a regression model can never reduce \(R^2\), so we should include all available predictor variables in the model.” Comment.

Exercise 2 (7 points)

Why is it not meaningful to attach a sign to the coefficient of multiple correlation \(R\), although we do so for the coefficient of simple correlation \(r_{12}\)?

Exercise 3 (7 points)

In a small-scale regression study, the following data were obtained,

Y \(X_1\) \(X_2\)
42.0 7.0 33.0
33.0 4.0 41.0
75.0 16.0 7.0
28.0 3.0 49.0
91.0 21.0 5.0
55.0 8.0 31.0

Assume the standard multiple regression model with independent normal error terms. Compute \(\mathbf{b}\), \(\mathbf{e}\), \(\mathbf{H}\), SSErr, \(R^2\), \(s^2_{\mathbf{b}}\), \(\hat Y\) for \(X_1 = 10\), \(X_2 = 30\). You can do the computations using software or by hand, although it would be lengthy to do them by hand.

Exercise 4 (14 points)

The data set “Brand preference” is available here.

It was collected to study thee relation between degree of brand liking (Y, first column) and moisture content (\(X_1\), second column) and sweetness (\(X_2\), third column) of the product.

  1. Fit a regression model to these data and state the estimated regression function. Interpret \(b_1\).
  2. Obtain residual plots. What information do they provide? Plot residuals against fitted values, against each predictor, and against the product of predictors.
  3. Test homoscedasticity.
  4. Conduct a formal lack of fit test.
  5. test whether the proposed linear regression model is significant. What do the results of the ANOVA F-test imply about the slopes?
  6. Estimate both slopes simultaneously using the Bonferroni procedure with at least a 99 percent confidence level.
  7. Report \(R^2\) and adjusted \(R^2\). How are they interpreted here?
  8. Calculate the squared correlation coefficient between \(Y_i\) and \(\hat Y_i\). Compare with part g).
  9. Obtain a 99% confidence interval for \(E(Y)\) when \(X_1 = 5\) and \(X_2 = 4\). Interpret it.
  10. obtain a 99% prediction interval for a new observation \(Y\) when \(X_1 = 5\) ane \(X_2 = 4\).