The article “Withdrawal Strength of Threaded Nails” describes an experiment comparing the ultimate withdrawal strengths (in N/mm) for several types of nails. For an annularly threaded nail with shank diameter 3.76 mm driven into spruce-pine-fir lumber, the ultimate withdrawal strength was modeled as lognormal with µ = 3.82 and ? = 0.219. For a helically threaded nail under the same conditions, the strength was modeled as lognormal with µ = 3.47 and ? = 0.272.

Question

a. What is the mean withdrawal strength for annularly threaded nails?
b. What is the mean withdrawal strength for helically threaded nails?
c. For which type of nail is it more probable that the withdrawal strength will be greater than 50 N/mm?
d. What is the probability that a helically threaded nail will have a greater withdrawal strength than the median for annularly threaded nails?
e. An experiment is performed in which withdrawal strengths are measured for several nails of both types. One nail is recorded as having a withdrawal strength of 20 N/mm, but its type is not given. Do you think it was an annularly threaded nail or a helically threaded nail? Why? How sure are you?

Answer 1

Let X represent the withdrawal strength for a randomly chosen annularly threaded nail, and let Y represent the withdrawal strength for a randomly chosen helically threaded nail.



(a)



(b)



(c) First find the probability for annularly threaded nails.



The z-score of 3.9120 is (3.9120 ? 3.82)/0.219 = 0.42.

The area to the right of z= 0.42 is 1 ? 0.6628 = 0.3372.

Therefore the probability for annularly threaded nails is 0.3372.

Now find the probability for helically threaded nails.

P(Y>50) = P(lnY>ln 50) = P(lnY>3.9120).

The z-score of 3.9120 is (3.9120 ? 3.47)/0.272 = 1.63.

The area to the right of z= 1.63 is 1 ? 0.9484 = 0.0516.

Therefore the probability for helically threaded nails is 0.0516.

Annularly threaded nails have the greater probability. The probability is 0.3372 vs. 0.0516 for helically threaded nails.

(d) First find the median strength for annularly threaded nails.

Let mbe the median of X. Then P(X? m) = P(lnX? lnm) = 0.5.

Since lnX~N(3.82, 0.2192), P(lnY<3.82) = 0.5.

Therefore lnm= 3.82, so m= e3.82.

Now

The z-score of 3.82 is (3.82 ? 3.47)/0.272 = 1.29.

The area to the right of z= 1.29 is 1 ? 0.9015 = 0.0985. Therefore the probability is 0.0985.



(e) The log of the withdrawal strength for this nail is ln 20 = 2.996.

For annularly threaded nails, the z-score of 2.996 is (2.996 ? 3.82)/0.219 = ?3.76.

The area to the left of z= ?3.76 is less than 0.0001.

Therefore a strength of 20 is extremely small for an annularly threaded nail; less than one in ten thousand suchnails have strengths this low.

For helically threaded nails, the z-score of 2.996 is (2.996 ? 3.47)/0.272 = ?1.74.

The area to the left of z= ?1.74 is 0.0409.

Therefore about 4.09% of helically threaded nails have strengths of 20 or below.

We can be pretty sure that it was a helically threaded nail. Only about 0.01% of annularly threaded nails have strengths as small as 20, while about 4.09% of helically threaded nails do.