Question

1) Steel rods are manufactured with a mean length of 20 cm. Because of variability in the manufacturing process, the lengths of the rods are approximately normally distributed, with standard deviation 1 cm. a. What proportion of rods has length less than 18.5 cm? b. What proportion of rods has length more than 22.5 cm? C. What proportion of rods has length between 18 cm to 22 cm? d. Any roads that is shorter than 16.98 cm and longer than 23.12 cm is discarded. What proportion of the rod is discarded?

Answer 1

Mean = 20

SD = 1

Z = (X - mean) / SD

A) P(X < 18.5) = P(Z < (18.5 - 20)/1)

= P(Z < -1.5)

= 0.0668

B) P(X > 22.5) = P(Z > (22.5 - 20)/1)

= P(Z > 2.5)

= 0.0062

C) P(18 < X < 22) = P((18-20)/1 < Z < (22-20)/1)

= P(-2 < Z < 2)

= P(Z < 2) - P(Z < -2)

= 0.9772 - 0.0228

= 0.9544

D) P(rod is discarded) = 1 - P(rod is not discarded)

= 1 - P(16.98 < X < 23.12)

= 1 - P((16.98-20)/1 < Z < (23.12-20)/1)

= 1 - (P(-3.02 < Z < 3.12))

= 1 - (P(Z < 3.12) - P(Z < -3.02))

= 1 - (0.9991 - 0.0013)

= 0.0022