Question

A company produces steel rods. The lengths of the steel rods are normally distributed with a mean of 92.6-cm and a standard deviation of 2-cm. For shipment, 12 steel rods are bundled together. Find the probability that the average length of a randomly selected bundle of steel rods is between 92-cm and 92.4-cm. Enter your answer as a number accurate to 4 decimal places.

Answer 1

Answer: 0.2140

Step-by-step explanation:

Given : A company produces steel rods. The lengths of the steel rods are normally distributed with

$\mu=92.6 \text{ cm}$

$\sigma=2\text{ cm}$

Sample size : $n=12$

Let x be the length of randomly selected item.

z-score : $z=\dfrac{x-\mu}{\dfrac{\sigma}{\sqrt{n}}}$

For x=92 cm

$z=\dfrac{92-92.6}{\dfrac{2}{\sqrt{12}}}\approx-1.04$

For x=92.4 cm

$z=\dfrac{92.4-92.6}{\dfrac{2}{\sqrt{12}}}\approx-0.35$

The probability that the average length of a randomly selected bundle of steel rods is between 92-cm and 92.4-cm by using the standard normal distribution table

= $P(92$

$= 0.3631693-0.14917=0.2139993\approx0.2140$

Hence, the probability that the average length of a randomly selected bundle of steel rods is between 92-cm and 92.4-cm is 0.2140.