A manufacturer claims that the tensile strength of a certain composite (in MPa) has the lognormal distribution with µ = 5 and ? = 0.5. Let X be the strength of a randomly sampled specimen of this composite.
a. If the claim is true, what is P(X <20)?
b. Based on the answer to part (a), if the claim is true, would a strength of 20 MPa be unusually small?
c. If you observed a tensile strength of 20 MPa, would this be convincing evidence that the claim is false? Explain.
d. If the claim is true, what is P(X< 130)?
e. Based on the answer to part (d), if the claim is true, would a strength of 130 MPa be unusually small?
f. If you observed a tensile strength of 130 MPa, would this be convincing evidence that the claim is false? Explain.
(a) P (X < 20) = P (ln X < ln20) = P (ln X < 2.996)
The z-score of 2.996 is (2.996 ? 5)/0.5 = ?4.01.
The area to the left of z = ?4.01 is negligible.
Therefore P(X <20) ? 0.
(b) Yes. Almost none of the specimens will have tensile strengths less than 20 MPa if the claim is true.
(c) Yes, because 20 MPa is unusually small if the claim is true.
(d) P (X<130) = P (ln X < ln130) = P(ln X < 4.868).
The z-score of 4.868 is (4.868 ? 5)/0.5 = ?0.26.
The area to the left of z = ?0.26 is 0.3974.
Therefore P(X <130) = 0.3974.
(e) No. Nearly 40% of the specimens will have tensile strengths less than 130 MPa if the claim is true.
(f) No, because 130 MPa is not unusually small if the claim is true.
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