An investigation of a die-casting process resulted in the accompanying data on x

Question

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.

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

(a) Carry out the model utility test.
State the appropriate hypotheses.

H0: ?1 = ?2 = 0
Ha: no ?i = 0

H0: ?1 = ?2 = 0
Ha: at least one ?i ? 0    

H0: ?1 ? ?2 ? 0
Ha: at least one ?i = 0

H0: ?1 ? ?2 ? 0
Ha: all ?i = 0

State the appropriate test statistic to two decimal places.


State the conclusion in the problem context.

Fail to reject H0. The model is not useful.

Reject H0. The model is not useful.    

Reject H0. The model is useful.

Fail to reject H0. The model is useful.

(b) Calculate a 95% confidence interval for ?2, the population regression coefficient of x2. (Round your answers to three decimal places.)

  ,

Interpret the confidence interval.

We estimate that the average change in temperature difference on the die surface will fall within this interval.

Holding both factors fixed, we estimate that the average change in temperature difference on the die surface will fall within this interval.   

Holding furnace temperature fixed, we estimate that the average change in temperature difference on the die surface will fall within this interval.

Holding die close time fixed, we estimate that the average change in temperature difference on the die surface will fall within this interval.

x1     1250 1300 1350 1250 1300 1250 1300 1350 1350 x2     6 7 6 7 6 8 8 7 8 y 80 95 101 85 92 87 96 106 108 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. Minitab output from fitting the multiple regression model with predictors x1 and x2 is given here. (a) Carry out the model utility test. State the appropriate hypotheses. State the appropriate test statistic to two decimal places. State the conclusion in the problem context. Fail to reject H0. The model is not useful. Reject H0. The model is not useful. Reject H0. The model is useful. Fail to reject H0. The model is useful. (b) Calculate a 95% confidence interval for ?2, the population regression coefficient of x2. (Round your answers to three decimal places.) Interpret the confidence interval. We estimate that the average change in temperature difference on the die surface will fall within this interval. Holding both factors fixed, we estimate that the average change in temperature difference on the die surface will fall within this interval. Holding furnace temperature fixed, we estimate that the average change in temperature difference on the die surface will fall within this interval. Holding die close time fixed, we estimate that the average change in temperature difference on the die surface will fall within this interval.

Explanation / Answer

(a)

H0: b1 = b2 = 0
Ha: at least one ?i ? 0

test statistic:

F=319.31

Reject H0. The model is useful.

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(b) (1.943, 4.057)

Holding furnace temperature fixed, we estimate that the average change in temperature difference on the die surface will fall within this interval.

Regression output confidence interval variables coefficients std. error    t (df=6) p-value 95% lower 95% upper Intercept -199.5556 11.6406 -17.143 2.52E-06 -228.0390 -171.0721 x1     0.2100 0.0086 24.299 3.19E-07 0.1889 0.2311 x2     3.0000 0.4321 6.943 .0004 1.9426 4.0574

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