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Suppose that a full 2 4 factorial uses the following factor:

Factor low (-) high (+)

Factor low (-) High (+)

A: Acid strength (%) 85 95

B: Reaction time (min) 15 35

C: Amount of acid (mL) 35 45

D: Reaction temp (degrees C) 60 80

The fitted model from this experiment is ŷ = 24 + 16x1 − 34x2 + 12x3 + 6x4 − 10x1x2 + 16x1x3. Predict

the response at the following points:

(a) A = 89, B = 20, C = 38, D = 66

(b) A = 90, B = 16, C = 40, D = 70

(c) A = 87, B = 28, C = 42, D = 61

(d) A = 90, B = 27, C = 37, D = 69

Answer :

The predicted responses for the given points are as follows:

(a) ŷ = 24 + 16(89) - 34(20) + 12(38) + 6(66) - 10(89)(20) + 16(89)(38).

(b) ŷ = 24 + 16(90) - 34(16) + 12(40) + 6(70) - 10(90)(16) + 16(90)(40).

(c) ŷ = 24 + 16(87) - 34(28) + 12(42) + 6(61) - 10(87)(28) + 16(87)(42).

(d) ŷ = 24 + 16(90) - 34(27) + 12(37) + 6(6) - 10(90)(27) + 16(90)(37).

In a 2^4 factorial experiment, four factors (A, B, C, D) are considered at two levels each, resulting in a total of 16 experimental runs. The fitted model ŷ represents the predicted response (output variable) based on the given factors and their interactions. Each coefficient in the model represents the effect of the corresponding factor or interaction.

To predict the response at specific points, we substitute the values of the factors into the fitted model. For example, to predict the response at point (a) with A = 89, B = 20, C = 38, and D = 66, we substitute these values into the model: ŷ = 24 + 16(89) - 34(20) + 12(38) + 6(66) - 10(89)(20) + 16(89)(38). Similarly, we can substitute the values for points (b), (c), and (d) to obtain the predicted responses.

By calculating these expressions, we can determine the predicted response values for each combination of factors, allowing us to evaluate the impact of the factors and their interactions on the response variable in the factorial experiment.

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