Question

The diameters of bolts produced in a machine shop are normally distributed with a mean of 5.11 millimeters and a standard deviation of 0.07 millimeters. Find the two diameters that separate the top 9% and the bottom 9%. These diameters could serve as limits used to identify which bolts should be rejected. Round your answer to the nearest hundredth, if necessary.

Answer 1

Answer:

The diameter that separates the bottom 9% is 5.02 millimeters.

The diameter that separates the top 9% is 5.2 millimeters.

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 5.11 millimeters and a standard deviation of 0.07 millimeters.

This means that $\mu = 5.11, \sigma = 0.07$

Bottom 9%:

The 9th percentile, which is X when Z has a pvalue of 0.09. So X when Z = -1.34.

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

$-1.34 = \frac{X - 5.11}{0.07}$

$X - 5.11 = -1.34*0.07$

$X = 5.02$

The diameter that separates the bottom 9% is 5.02 millimeters.

Top 9%:

The 100 - 9 = 91th percentile, which is X when Z has a pvalue of 0.91. So X when Z = 1.34.

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

$1.34 = \frac{X - 5.11}{0.07}$

$X - 5.11 = 1.34*0.07$

$X = 5.2$

The diameter that separates the top 9% is 5.2 millimeters.