Question

Tensile strength tests were carried out on two different grades of wire rod, resulting in the accompanying data.

Grade	Sample Size	Sample Mean (kg/mm^2)	sample SD
AISI 1064	m=126	x bar = 102.1	s1 = 1.2
AISI 1078	n = 126	y bar = 128.5	s2 = 2.0

(a) Does the data provide compelling evidence for concluding that true average strength for the 1078 grade exceeds that for the 1064 grade by more than 10 kg/mm²? Test the appropriate hypotheses using a significance level of 0.01. State the relevant hypotheses.

H₀: μ₁₀₆₄ − μ₁₀₇₈ = −10
H_a: μ₁₀₆₄ − μ₁₀₇₈ ≤ −10

H₀: μ₁₀₆₄ − μ₁₀₇₈ = −10
H_a: μ₁₀₆₄ − μ₁₀₇₈ ≥ −10    

H₀: μ₁₀₆₄ − μ₁₀₇₈ = −10
H_a: μ₁₀₆₄ − μ₁₀₇₈ ≠ −10

H₀: μ₁₀₆₄ − μ₁₀₇₈ = −10
H_a: μ₁₀₆₄ − μ₁₀₇₈ < −10

H₀: μ₁₀₆₄ − μ₁₀₇₈ = −10
H_a: μ₁₀₆₄ − μ₁₀₇₈ > −10

Calculate the test statistic and 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.

Reject H₀. The data suggests that the true average strength for the 1078 grade exceeds that for the 1064 grade by more than 10 kg/mm².

Fail to reject H₀. The data does not suggest that the true average strength for the 1078 grade exceeds that for the 1064 grade by more than 10 kg/mm².    

Reject H₀. The data does not suggest that the true average strength for the 1078 grade exceeds that for the 1064 grade by more than 10 kg/mm².

Fail to reject H₀. The data suggests that the true average strength for the 1078 grade exceeds that for the 1064 grade by more than 10 kg/mm².

(b) Estimate the difference between true average strengths for the two grades in a way that provides information about precision and reliability. (Use a 95% confidence interval. Round your answers to two decimal places.)

(_,_) kg/mm²

Answer 1

The statistical software output for this problem is:

Hence,

a) H₀: μ₁₀₆₄ − μ₁₀₇₈ = −10
H_a: μ₁₀₆₄ − μ₁₀₇₈ < −10

z = -78.93

p - Value = 0.0000

Conclusion: Reject H₀. The data suggests that the true average strength for the 1078 grade exceeds that for the 1064 grade by more than 10 kg/mm².

b) 95% confidence interval:

(-26.81, -25.99)