Question

An experiment was performed to compare the fracture toughness of high-purity 18 Ni maraging steel with commercial-purity steel of the same type. For m = 31 specimens, the sample average toughness was x = 63.4 for the high-purity steel, whereas for n = 37 specimens of commercial steel y = 57.9. Because the high-purity steel is more expensive, its use for a certain application can be justified only if its fracture toughness exceeds that of commercial-purity steel by more than 5. Suppose that both toughness distributions are normal.

(a) Assuming that σ₁ = 1.3 and σ₂ = 1.1, test the relevant hypotheses using α = 0.001. (Use μ₁ − μ₂, where μ₁ is the average toughness for high-purity steel and μ₂ is the average toughness for commercial steel.)
State the relevant hypotheses. (Choose of the following)

1. H₀: μ₁ − μ₂ = 5
H_a: μ₁ − μ₂ > 5

2. H₀: μ₁ − μ₂ = 5
H_a: μ₁ − μ₂ < 5   

3. H₀: μ₁ − μ₂ = 5
   H_a: μ₁ − μ₂ ≤ 5

4. H₀: μ₁ − μ₂ = 5
H_a: μ₁ − μ₂ ≠ 5

Calculate the test statistic and determine the P-value. (Round your test statistic to two decimal places and your P-value to four decimal places.)

z	=
P-value	=

State the conclusion in the problem context. (Choose of the following)

1. Fail to reject H₀. The data does not suggest that the fracture toughness of high-purity steel exceeds that of commercial-purity steel by more than 5.

2. Reject H₀. The data suggests that the fracture toughness of high-purity steel exceeds that of commercial-purity steel by more than 5.     

3. Reject H₀. The data does not suggest that the fracture toughness of high-purity steel exceeds that of commercial-purity steel by more than 5.

4. Fail to reject H₀. The data suggests that the fracture toughness of high-purity steel exceeds that of commercial-purity steel by more than 5.

(b) Compute β for the test conducted in part (a) when μ₁ − μ₂ = 6. (Round your answer to four decimal places.)
β = __________

Answer 1