Question

A company produces steel rods. The lengths of the steel rods are normally distributed with a mean of 106.4-cm and a standard deviation of 1.8-cm. For shipment, 9 steel rods are bundled together. Round all answers to four decimal places if necessary.

A. What is the distribution of X? X ~ N(     ,     )

B. What is the distribution of ¯x? ¯x ~ N(      ,    )

C. For a single randomly selected steel rod, find the probability that the length is between 105.9-cm and 106.5-cm.

D. For a bundled of 9 rods, find the probability that the average length is between 105.9-cm and 106.5-cm.

E. For part d), is the assumption of normal necessary?   Yes or No

Answer 1

Let ,

A. The distribution of X is ,

B. If , then

Where , and

C. Now ,

; From standard normal probability table

Therefore , for a single randomly selected steel rod, the probability that the length is between 105.9-cm and 106.5-cm is 0.1342.

D) Now ,

; From standard normal probability table

Therefore , for a bundled of 9 rods, the probability that the average length is between 105.9-cm and 106.5-cm is 0.3642.

E) Yes, the assumption of normality is necessary for part D.