An investigation of a die-casting process resulted in the accompanying data on x₁=  furnace temperature , x₂ = die close time, and y = temperature difference on the die surface (“A Multiple-Objective Decision-Making Approach for Assessing Simultaneous Improvement in Die Life and Casting Quality in a Die Casting Process,” Quality Engineering, 1994: 371–383).

  

      

  

Minitab output from fitting the multiple regression model with predictors x₁ and x₂ is given here.

  

      

a. Carry out the model utility test.

b. Calculate and interpret a 95% confidence interval for β₂, the population regression coefficient of x₂.

c. When x₁ = 1300 and x₂ = 7 , the estimated standard deviation of Calculate a 95% confidence interval for true average temperature difference when furnace temperature is 1300 and die close time is 7.

d. Calculate a 95% prediction interval for the temperature difference resulting from a single experimental run with a furnace temperature of 1300 and a die close time of 7.

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X = x2 X1 x2 y X1 x2 y 1250 6 80 1250 8 87 1300 7 95 1300 8 96 s = 1.058 Predictor Coef Constant -199.56 furntemp 0.210000 clostime 3.0000 R-sq 1350 6 101 Minitab output from fitting the multiple regression model with predictors x, and x₂ is given here. The regression equation is tempdiff = -200+ 0.210 furntemp +3.00 clostime = 1350 7 106 99.1% 1250 7 85 1350 8 108 0.008642 0.4321 Stdev t-ratio 11.64 -17.14 24.30 6.94 1300 6 92 = р 0.000 0.000 0.000 R-sq (adj) 98.8% X = x2 X1 x2 y X1 x2 y 1250 6 80 1250 8 87 1300 7 95 1300 8 96 s = 1.058 Predictor Coef Constant -199.56 furntemp 0.210000 clostime 3.0000 R-sq 1350 6 101 Minitab output from fitting the multiple regression model with predictors x, and x₂ is given here. The regression equation is tempdiff = -200+ 0.210 furntemp +3.00 clostime = 1350 7 106 99.1% 1250 7 85 1350 8 108 0.008642 0.4321 Stdev t-ratio 11.64 -17.14 24.30 6.94 1300 6 92 = р 0.000 0.000 0.000 R-sq (adj) 98.8%