Question

A company produces steel rods. The lengths of the steel rods are normally distributed with a mean of 222-cm and a standard deviation of 1.5-cm. For shipment, 9 steel rods are bundled together.

Find the probability that the average length of a randomly selected bundle of steel rods is less than 222.1-cm.

P(M < 222.1-cm) =

Enter your answer as a number accurate to 4 decimal places. Answers obtained using exact z-scores or z-scores rounded to 3 decimal places are accepted.

Answer 1

A company produces steel rods.

The length of the steel rods are normally distributed with a mean of 222 cm and a standard deviation of 1.5 cm.

For shipment, 9 steel rods are bundled together.

Now, if X be the length of a randomly selected rod, then X follows normal with mean 222 cm and standard deviation of 1.5 cm.

Now, if M be the average length of a randomly selected bundle, then M will also follow normal distribution.

We note,

So, we can say that the average length M of a bundle, follows normal distribution with mean 222 and standard deviation of 0.5.

This means,

Z=(M-222)/0.5 follows standard normal with mean 0 and standard deviation of 1.

We have to find

Where, Z is the standard normal variate.

Where, phi is the distribution function of the standard normal variate.

From the standard normal table, this becomes

The answer is

The probability, that the average length of a randomly selected bundle of steel rods is less than 222.1 cm, is 0.5793.