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

Answer:

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.

Step-by-step 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:

$Z = \frac{X - \mu}{\sigma}$

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.

$Z = \frac{X - \mu}{\sigma}$

$1.405 = \frac{X - 5.16}{0.04}$

$X - 5.16 = 1.405*0.04$

$X = 5.2162$

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.

$Z = \frac{X - \mu}{\sigma}$

$-1.405 = \frac{X - 5.16}{0.04}$

$X - 5.16 = -1.405*0.04$

$X = 5.1038$

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.