Question

The diameters of bolts produced in a machine shop are normally distributed with a mean of 5.7 millimeters and a standard deviation of 0.08 millimeters. Find the two diameters that separate the top 3% and the bottom 3%. 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

The diameter that separates the top 3% is of 5.85 millimeters, and the one which separates the bottom 3% is of 5.55 millimeters.

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 5.7 millimeters and a standard deviation of 0.08 millimeters.

This means that
`\mu = 5.7, \sigma = 0.08`

Top 3%

The 100 - 3 = 97th percentile, which is X when Z has a p-value of 0.97, so X when Z = 1.88.

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

`1.88 = (X - 5.7)/(0.08)`

`X - 5.7 = 1.88*0.08`

`X = 5.85`

Bottom 3%

The 3rd percentile, which is X when Z has a p-value of 0.03, so X when Z = -1.88.

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

`-1.88 = (X - 5.7)/(0.08)`

`X - 5.7 = -1.88*0.08`

`X = 5.55`

The diameter that separates the top 3% is of 5.85 millimeters, and the one which separates the bottom 3% is of 5.55 millimeters.