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

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=(x-\mu)/((\sigma)/(√(n)))`

For x=92 cm

`z=(92-92.6)/((2)/(√(12)))\approx-1.04`

For x=92.4 cm

`z=(92.4-92.6)/((2)/(√(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<x<92.4)=P(-1.04<z<-0.35)=P(z<-0.35)-P(z<-1.04)`

`= 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.