27. New Construction. Our construction firm is interested in forecasting new construction in the United States for the years 2002 and 2003. We have data in billions of dollars for the years 1991 through 2001 from the Department of Commerce. These data are in the file NEWCON3 on the CD and are shown in Table $3.8$.

1. Fit a linear trend to these data. What is the resulting regression equation?
2. What percentage of the variation in \(y\) has been explained by the regression?
3. Based on your answer in part b and on any other regression results you obtain, how well does the equation fit the data? Does a good fit ensure that forecasts for future years will be accurate?
4. Use the equation developed to predict new construction in both 2002 and 2003 . Find a point prediction and a \(95 \%\) prediction interval.
5. How reliable do you believe the forecast in part d might be? What factors might influence this accuracy?

29. Wheat Exports. The relationship between exthange rates and agricultural exports is of interest to agricultural economists. One such export of merest is wheat. The following data

\(y\), U.S. wheat export shipments (SHIPMENT)

\(x\), the real index of weighted-average exchange rates for the U.S. dollar (EXCHRATE)

are available in a file named WHEAT3 on the CD.

These time-series data were observed monthly from January 1974 through March 1985. Perform any analyses necessary to answer the following questions:

1. What is the estimated regression equation relating \(y\) to \(x\) ?
2. Are \(y\) and \(x\) linearly related? Conduct a hypothesis test to answer this question and use a \(5 \%\) level of significance. State the hypotheses to be tested, the decision rule, the test statistic, and your decision. What conclusion can be drawn from the result of the test?
3. What percentage of the variation in \(y\) has been explained by the regression?
4. Construct a \(95 \%\) confidence interval estimate of \(\beta_{1}\).

32. Major League Baseball Wins. What factor is most important in building a winning baseball team? Some might argue for a high batting ave age. Or it might be a team that hits for power a measured by the number of home runs. On the other hand, many believe that it is quality pitch' as measured by the earned run average of the team's pitchers. The file MLB3 on the CD costains data on the following variables for the 30 major league baseball teams during the 2002 season:

\(\begin{aligned}

\text { WINS } &=\text { number of games won } \\

\mathrm{HR} &=\text { number of home runs hit } \\

\mathrm{BA} &=\text { average batting average } \\

\text { ERA } &=\text { earned run average }

\end{aligned}\)

Using WINS as the dependent variable, use scatterplots and regression to investigate the relationship of the other three variables to WINS. Which of the three possible explanatory

Hiables exhibits the strongest relationship to WN\$' What might this suggest to managers of inicion league baseball teams?

34. Fanfare. Fanfare International, Inc., designs, distributes, and markets ceiling fans and lighting fixtures. The company's product line includes 120

basic models of ceiling fans and 138 compatible fan light kits and table lamps. These products are marketed to over 1000 lighting showrooms and electrical wholesalers that supply the remodeling and new construction markets. The product line is distributed by a sales organization of 58 independent sales representatives.

In the summer of 1994 , Fanfare decided it needed to develop forecasts of future sales to help determine future salesforce needs, capital expenditures, and so on. The file named FAN3 on the CD contains monthly sales data and data on three additional variables for the period July 1990 through May 1994. The variables are defined as follows

SALES = total monthly sales in thousands of dollars

ADEX = advertising expense in thousands of dollars

MTGRATE = mortgage rate for 30 -year loans \((\%)\)

HSSTARTS = housing starts in thousands of units

The data file contains the four variables as shown, plus columns for year and month. The sales data have been transformed to provide confidentiality.

As a consultant to Fanfare, your job is to find the best single variable to forecast future sales. Try each of the three variables in a simple regression and decide which is the best to create a forecasting model for Fanfare. Justify your choice. What problems do you see with using each of the three possible variables to help forecast sales?

37. Cubs Attendance. The Chicago Cubs baseball organization is interested in examining the relationship between attendance and the number of wins during the season. One possible hypothesized model is

ATTENDANCE \(=\beta_{0}+\beta_{1}\) WINS \(+e\)

They plan to use the equation to forecast future attendance and have annual data from 1972 through 1999 . These data are shown in Table $3.10$ and are in the file named CUBSWIN3 on the CD.

1. Find the regression equation using ATTENDANCE as the dependent variable and WIS as the explanatory variable.
2. The Cubs organization wants to use the regression to forecast attendance next season: If we win 110 games next year, that means our forecast for attendance will be almost $2,600,000$. With Sammy Sosa back and if Kerry Wood is healthy, I think we've got a good chance at 110 wins. Do you see any problems with using the forecast for attendance specifically when the number of wins is 110 ?

1. Cost Control. Ms. Karen Ainsworth is an employee of a well-known accounting firm's management services division. She is currently on a consulting assignment to the Apex Corporation, a firm that produces corrugated paper for use in making boxes and other packing materials. Apex called in consulting help to improve its cost control program, and Ms. Ainsworth is analyzing manufacturing costs to understand more fully the important influences on these costs. She has assembled monthly data on a group of variables, and she is using regression analysis to help assess how these variables are related to total manufacturing cost. The variables Ms. Ainsworth has selected to study, the data for which are contained in the file COST4 on the CD, are

\(y\), total manufacturing cost per month in thousands of dollars (COST)

\(x_{1}\), total production of paper per month in tons (PAPER)

\(x_{2}\), total machine hours used per month (MACHINE)

\(x_{3}\), total variable overhead costs per month in thousands of dollars (OVERHEAD)

\(x_{4}\), total direct labor hours used each month (LABOR)

The data shown in Table $4.2$ refer to the period January 2001 through March 2003. Ms. Ainsworth wants to use a cost function developed by means of regression analysis that initially includes all four cif the explanatory variables. Use the regression stats in Figure $4.8$ to help answer the following question

1. What is the equation that is determined using all four explanatory variables?
2. Conduct the \(F\) test for overall fit of the regression. State the hypotheses to be tested, the decision rule, the test statistic, and your decision. \(\mathrm{L}=\) a \(5 \%\) level of significance. What conclusion cart be drawn from the result of the test?
3. In the cost accounting literature, the sample regression coefficient corresponding to \(x_{k}\) is regarded as an estimate of the true marginal cost of output associated with the variable \(x\). Find a point estimate of the true marginal cos associated with total machine hours per month. Also, find a \(95 \%\) confidence interval estimate of the true marginal cost associated with to the machine hours.
4. Test the hypothesis that the true marginal cost of output associated with total production of paper is $1.0$. Use a \(5 \%\) level of significance and a two-tailed test procedure. State the hypotheses to be tested, the decision rule, the test statistic, and your decision. What conclusion can be drawn from the result of the test
5. What percentage of the variation in \(y\) has been explained by the regression?

[Table 4.2 Data and Figure 4.8 Regression Results are on next page.]

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