91Ó°ÊÓ

Grocery retailer. A large, national grocery retailer tracks productivity and costs of \(\frac{k}{\text { fts facilities }}\) closely, Data below were obtained from a single distribution center for a one-year period, Each data point for each variable represents one week of activity. The variables included are the number of cases shipped \(\left(X_{1}\right),\) the indirect costs of the total labor hours as a percentage \(\left(X_{2}\right)\) a qualitative predictor called holiday that is coded 1 if the week has a holiday and 0 otherwise \(\left(X_{3}\right),\) and the total labor hours \((Y)\) $$\begin{array}{cccccccc} i: & 1 & 2 & 3 & \dots & 50 & 51 & 52 \\ \hline \chi_{n}: & 305,657 & 328,476 & 317,164 & \dots & 290,455 & 411,750 & 292,087 \\ \chi_{i 2}: & 7.17 & 6.20 & 4.61 & \dots & 7.99 & 7.83 & 7.777 \\ x_{13}: & 0 & 0 & 0 & \dots & 0 & 0 & 0 \\ y_{i}: & 4264 & 4496 & 4317 & \dots & 4499 & 4186 & 4342 \end{array}$$ a. Prepare separate stem-and-leaf plots for the number of cases shipped \(X_{i 1}\) and the indirect cost of the total hours \(X_{i 2}\). Are there any outlying cases present? Are there any gaps in the data? b. The cases are given in consecutive weeks. Prepare a time plot for cach predictor variable. What do the plots show? c. Obtain the scatter plot matrix and the correlation matrix. What information do these diagnostic aids provide here?

For each of the following regression models, indicate whether it is a general linear regression model. If it is not, state whether it can be expressed in the form of (6.7) by a suitable transformation: a. \(Y_{i}=\beta_{0}+\beta_{1} X_{i 1}+\beta_{2} \log _{10} X_{i 2}+\beta_{3} X_{i 1}^{2}+\varepsilon_{i}\) b. \(Y_{i}=\varepsilon_{i} \exp \left(\beta_{0}+\beta_{1} X_{n}+\beta_{2} X_{i 2}^{2}\right)\) c. \(Y_{i}=\log _{10}\left(\beta_{1} X_{i 1}\right)+\beta_{2} X_{i 2}+\varepsilon_{i}\) d. \(Y_{i}=\beta_{0} \exp \left(\beta_{1} X_{11}\right)+\varepsilon_{i}\) e. \(Y_{i}=\left[1+\exp \left(\beta_{0}+\beta_{1} X_{i 1}+\varepsilon_{i}\right)\right]^{-1}\)

(Calculus needed.) Consider the multiple regression model: $$Y_{i}=\beta_{1} X_{i 1}+\beta_{2} X_{i 2}+\varepsilon_{i} \quad i=1, \ldots, n$$ where the \(\varepsilon_{i}\) are uncorrelated, with \(E\left[\varepsilon_{i}\right]=0\) and \(\sigma^{2}\left\\{\varepsilon_{i}\right\\}=\sigma^{2}\) a. 'State the least squares criterion and derive the least squares estimators of \(\beta_{1}\) and \(\beta_{2}\). b. Assuming that the \(\varepsilon_{i}\) are independent normal random variables, state the likelihood function and obtain the maximum likelihood estimators of \(\beta_{1}\) and \(\beta_{2}\). Are these the same as the least squares estimators?

In a small-scale regression study, the following data were obtained: $$\begin{array}{crrrrrr} i & 1 & 2 & 3 & 4 & 5 & 6 \\ \hline x_{n}: & 7 & 4 & 16 & 3 & 21 & 8 \\ X_{i 2}: & 33 & 47 & 7 & 49 & 5 & 31 \\ Y_{i}: & 42 & 33 & 75 & 28 & 91 & 55 \end{array}$$ Assume that regression model ( 6.1 ) with independent normal error terms is approptiate. Using matrix methods, obtain \((a) \mathbf{b} ;(b) e_{i}(c) H ;(d) S S R ;(e) s^{2}(b) ;(f) \hat{Y}_{h}\) when \(X_{h 1}=10, X_{h 2}=30$$(g) s^{2}\left(\hat{Y}_{h}\right)\) when \(X_{h 1}=10, X_{h 2}=30\)