An investigation of a diecasting process resulted in the acc

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
H_a: no ?_i = 0

H₀: ?₁ = ?₂ = 0
H_a: at least one ?_i ? 0    

H₀: ?₁ ? ?₂ ? 0
H_a: at least one ?_i = 0

H₀: ?₁ ? ?₂ ? 0
H_a: 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.

Solution

(a)

H₀: b₁ = _b2 = 0
H_a: 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