1. Yield strength of steel alloy.  Industrial engineers at the University of Florida used regression modeling as a tool to reduce the time and cost associated with developing new metallic alloys (Modelling and Simulation in Materials Science and Engineering, Vol. 13, 2005).  To illustrate, the engineers built a regression model for the tensile yield strength (y) of a new steel alloy.  The potential important predictors of yield strength are listed below.

x₁ = Carbon amount (% weight)

x₂ = Manganese amount (% weight)

x₃ = Chromuim amount (% weight)

x₄ = Nickel amount (% weight)

x₅ = Molybdenum amount (% weight)

x₆ = Copper amount (% weight)

x₇ = Nitrogen amount (% weight)

x₈ = Vanadium amount (% weight)

x₉ = Plate thickness (millimeters)

x₁₀ = Solution treating (millimeters)

x₁₁ = Aging temperature (degrees,Celsius)

a. The engineers used stepwise regression in order to search for a parsimonious set of predictor variables.  Do you agree with the decision? Explain.

b. The stepwise regression selected the following independent variables:

x_l = Carbon, x₂ = Manganese, x₃ = Chromium, x₅ = Molybdenum, x₆ = Copper, x₈ = Vanadium, x₉ = Plate thickness, x₁₀ = Solution treating, and x₁₁ = Aging temperature. Based on this information, determine the total number of first - order models that were fit in the stepwise routine.

c. Refer to b.  All these variables were statistically significant in the stepwise model to predict yield strength.  Do you agree with this decision? Explain.

2. Women in top management. The Journal of Organizational Culture, Communications and Conflict (July 2007) published a study on women in upper-management positions at U.S. firms. Observational data (n=252 months) were collected for several variables in an attempt to model the number of females in managerial positions (y). The independent variables included the number of females with a college degree (_x1), the number of female high school graduates with no college degree (x₂), the number of males in managerial positions (x₃), the number of males with a college degree(x₄), and the number of male high school graduates with no college degree (x₅)

a. The correlation relating number of females in managerial positions and number of females with a college degree was determined to be r=.983. Can the researchers concludethat an increase in the number of females with a college degree will cause the number of
females in managerial positions to increase? Explain.

b. The correlation relating number of males in managerial positions and number of males with a college degree was determined to be r=.722. What potential problem can occur in the regression analysis? Explain.

3. Characteristics of sea ice melt ponds. Surface albedo is defined as the ratio of solar energy directed upward from a surface over energy incident upon the surface. Surface albedo is a critical climatological parameter of sea ice. The National Snow and Ice Data Center (NSIDC) collects data on the albedo, depth, and physical characteristics of ice melt ponds in the Canadian Arctic, including ice type (classified as first-year ice, multiyear ice, or landfast ice). Data for 504 ice melt ponds located in the Barrow Strait in the Canadian Arctic are saved in the PONDICE file. Environmental engineers want to model the broadband surface albedo level, y, of the ice as a function of pond depth, _x1 (meters), and ice type, represented by the dummy variables x₂={1 if first-year ice, 0 if not} and x₃={1 if multiyear ice, 0 if not}. Ultimately, the engineers will use the model to predict the surface albedo level of an ice melt pond. Access the data in the PONDICE file and identify the experimental region for the engineers. What advice do you give them about the use of the prediction equation?

4. FDA investigation of a meat-processing plant. A particular meat-processing plant slaughters steers and cuts and wraps the beef for its customers. Suppose a complaint has been filed with Food and Drug Administration (FDA) against the processing plant. The complaint alleges that the consumer does not get all the beef from the steer he purchases. In particular, one consumer purchased a 300-pound steer but received only 150 pounds of cut and wrapped beef. To settle the complaint, the FDA collected data on the live weights and dressed weights of nine steers processed by a reputable meat-processing plant (not the firm in question). The results are listed in the table.

STEERS

a. Fit the model E(y) = β₀ + β₁x to the data.

b. Construct a 95% prediction interval for the dressed weight y of a 300-pound steer.

c. Would you recommend that the FDA use the interval obtained in part b to determine whether the dressed weight of 150 pounds is a reasonable amount to receive from a 300-pound steer? Explain.