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Question: Yield strength of steel alloy. Industrial engineers at the University of Florida used regression modelling 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 potentially important predictors of yield strength are listed in the accompanying table.

a. The engineers discovered that the variable Nickel (x₄) was highly correlated with the other potential independent variables. Consequently, Nickel was dropped from the model. Do you agree with this decision? Explain.

b. The engineers used stepwise regression on the remaining 10 potential independent variables in order to search for a parsimonious set of predictor variables. Do you agree with this decision? Explain.

c. The stepwise regression selected the following independent variables: x₁ = Carbon, x₂ = Manganese, x₃ = Chromium, x₅ = Molybdenum, x₆ = Copper, x₈ = Vanadium, x₉ = Plate thickness, x₁₀ = Solution treating, and x₁₁ = Aging temperature. All these variables were statistically significant in the step-wise model, with R² = .94. Consequently, the engineers used the estimated stepwise model to predict yield strength. Do you agree with this decision? Explain.

Step by step solution

01

Given information

The variable Nickel was highly correlated with the other potential independent variables. Consequently, Nickel was dropped from the model. The stepwise regression model is used.

02

Multicollinearity

a.

Since Nickel (x₄) was highly correlated with other variables, it is prudent to remove the variable from the model. Otherwise, the model will not give us unbiased results due to the problem of multicollinearity.

03

Significance of stepwise regression

b.

When there are a lot of independent variables, it is important to only include statistically significant independent variables in the final regression model. Hence, it is always better to conduct a stepwise regression analysis and only include independent variables which are significant to fitting the data.

04

Stepwise regression 

c.

The stepwise regression selected following independent variables x₁ = Carbon, x₂ = Manganese, x₃ = Chromium, x₅ = Molybdenum, x₆ = Copper, x₈ = Vanadium, x₉ = Plate thickness, x₁₀ = Solution treating, and x₁₁ = Aging temperature.

Here, the value of R² is 0.94 indicating that 94% of the variation in the data can be explained using the model which shows that the model is an ideal fit for the data. The model produced by the stepwise regression method can be used for further calculation and analysis.

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