The lengths of nails produced in a factory are normally distributed with a mean of 6.02 6.02 centimeters and a standard deviation of 0.05 0.05 centimeters. Find the two lengths that separate the top 9% 9% and the bottom 9% 9%. These lengths could serve as limits used to identify which nails should be rejected. Round your answer to the nearest hundredth, if necessary.

Question

Antony Estrada

Mathematics

7 April, 14:58

The lengths of nails produced in a factory are normally distributed with a mean of 6.02 6.02 centimeters and a standard deviation of 0.05 0.05 centimeters. Find the two lengths that separate the top 9% 9% and the bottom 9% 9%. These lengths could serve as limits used to identify which nails should be rejected. Round your answer to the nearest hundredth, if necessary.

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Answers (1)

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Jakobe Chan · Answer 1

Step-by-step explanation:

First, use a z-score table or calculator to find the z-score that corresponds to the percentile. Using a calculator, z = - 1.3408 is the bottom 9%, and z = 1.3408 is the top 9%.

Now calculate the length that corresponds to these z scores.

z = (x - μ) / σ

-1.3408 = (x - 6.02) / 0.05

x = 5.95

1.3408 = (x - 6.02) / 0.05

x = 6.09

So the bottom 9% and the top 9% are between 5.95 cm and 6.09 cm.