Question

The lengths of nails produced in a factory are normally distributed with a mean of 4.84 centimeters and a standard deviation of 0.05 centimeters. Find the two lengths that separate the top 3% and the bottom 3%. These lengths could serve as limits used to identify which nails should be rejected. Round your answer to the nearest hundredth, if necessary.

Answer 1

Top 3%: 4.934 cm

Bottom 3%: 4.746 cm

Explanation:

Problems of normally distributed samples are 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 = 4.84, \sigma = 0.05`

Top 3%

Value of Z when Z has a pvalue of 1 - 0.03 = 0.97. So X when Z = 1.88.

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

`1.88 = (X - 4.84)/(0.05)`

`X - 4.84 = 0.05*1.88`

`X = 4.934`

Bottom 3%

Value of Z when Z has a pvalue of 0.03. So X when Z = -1.88.

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

`-1.88 = (X - 4.84)/(0.05)`

`X - 4.84 = 0.05*(-1.88)`

`X = 4.746`