The diameters of bolts produced in a machine shop are normally distributed with a mean of 5.74 millimeters and a standard deviation of 0.07 millimeters. Find the two diameters that separate the top 5% and the bottom 5%. These diameters could serve as limits used to identify which bolts should be rejected. Round your answer to the nearest hundredth, if necessary.

The value that separates the top 5% is 5.86 millimeters.

The value that separates the bottom 5% is 5.62 millimeters.

Step-by-step explanation:

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

In a set with mean and standard deviation , the zscore of a measure X is given by:

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:

Top 5%

value of X when Z has a pvalue of 1-0.05 = 0.95. So X when Z = 1.645.

The value that separates the top 5% is 5.86 millimeters.

Bottom 5%

value of X when Z has a pvalue of 0.05. So X when Z = -1.645

The value that separates the bottom 5% is 5.62 millimeters.

step-by-step explanation:

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