Question

The diameters of bolts produced in a machine shop are normally distributed with a mean of 5.26 millimeters and a standard deviation of 0.07 millimeters. Find the two diameters that separate the top 10% and the bottom 10%. 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

Top 10%: 5.35 millimeters

Bottom 10%: 5.17 millimeters

Explanation:

Problems of normally distributed samples can be solved using the z-score formula.

In a set with mean
`\mu` and standard deviation
`\sigma`, the zscore 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 pvalue, we get the probability that the value of the measure is greater than X.

In this problem, we have that:

`\mu = 5.26, \sigma = 0.07`

Top 10%

X when Z has a pvalue of 1-0.1 = 0.9. So X when Z = 1.28.

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

`1.28 = (X - 5.26)/(0.07)`

`X - 5.26 = 1.28*0.07`

`X = 5.35`

Bottom 10%

X when Z has a pvalue of 0.1. So X when Z = -1.28

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

`-1.28 = (X - 5.26)/(0.07)`

`X - 5.26 = -1.28*0.07`

`X = 5.17`