Question

The lengths of nails produced in a factory are normally distributed with a mean of 5.16 centimeters and a standard deviation of 0.04 centimeters. Find the two lengths that separate the top 8% and the bottom 8%.

Answer 1

The length that separates the top 8% is of 5.2162 centimeters, and the length that separates the bottom 8% is of 5.1038 centimeters.

Explanation:

Normal Probability Distribution

Problems of normal distributions can be solved using the z-score formula.

In a set with mean
`\mu` and standard deviation
`\sigma`, the z-score of a measure X is given by:

`Z = (X - \mu)/(\sigma)`

The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the p-value, we get the probability that the value of the measure is greater than X.

Mean of 5.16 centimeters and a standard deviation of 0.04 centimeters.

This means that
`\mu = 5.16, \sigma = 0.04`

Length that separates the top 8%

The 100 - 8 = 92th percentile, which is X when Z has a p-value of 0.92, so X when Z = 1.405.

`Z = (X - \mu)/(\sigma)`

`1.405 = (X - 5.16)/(0.04)`

`X - 5.16 = 1.405*0.04`

`X = 5.2162`

Length that separates the bottom 8%

This is the 8th percentile, which is X when Z has a p-value of 0.08, so X when Z = -1.405.

`Z = (X - \mu)/(\sigma)`

`-1.405 = (X - 5.16)/(0.04)`

`X - 5.16 = -1.405*0.04`

`X = 5.1038`

The length that separates the top 8% is of 5.2162 centimeters, and the length that separates the bottom 8% is of 5.1038 centimeters.