Specifications for an aircraft bolt require that the ultimate tensile strength be at least 18 kN. It is known that 10% of the bolts have strengths less than 18.3 kN and that 5% of the bolts have strengths greater than 19.76 kN. It is also known that the strengths of these bolts are normally distributed.
a. Find the mean and standard deviation of the strengths.
b. What proportion of the bolts meet the strength specification?
a) Since 10% of the bolts have strengths less than 18.3 kN, the z-score of 18.3 is ?
= ?1.28.
Since 5% of the bolts have strengths greater than 19.76 kN, the z-score of 19.76 is 
= 1.645.
Therefore (18.3 – ?)/? = ?1.28, and (19.76 ? ?)/? = 1.645.
Solving for ? and ? yields ? = 18.9389 and ? = 0.4991.
(b) z= (18?18.9389)/0.4991 = ?1.88. The area to the right of z = ?1.88 is 0.9699.
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