Question

Specifications for an aircraft bolt require that the ultimate tensile strength be at least 18 kN. It is known that 5% of the bolts have strength less than 18.5 kN and 10% of the bolts have strengths greater than 20.0 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?

Answer 1

a) Let the mean be and standard deviation be

P(X < 18.5) = 0.05

P(Z < (18.5 - )/) = 0.05

From standard normal distribution table, (18.5 - )/ = -1.645

= 18.5 + 1.645

P(X < 20) = 0.10

P(Z < (20 - )/) = 0.10

From standard normal distribution table, (20 - )/ = -1.28

= 20 + 1.28

18.5 + 1.645 = 20 + 1.28

= 4.11 kN

= 18.5 + 1.645 x 4.11

= 25.26 kN

b) Proportion of the bolts meet the strength specification = P(X 18)

= 1 - P(X < 18)

= 1 - P(Z < (18-25.26)/4.11)

= 1 - P(Z < -1.77)

= 1 - 0.0384

= 0.9616