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The lengths of nails produced in a factory are normally distributed with a mean of 6.13 centimeters and a standard deviation of 0.06 centimeters.Find the two lengths that separate the top 7% and the bottom 7%.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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Answer:

Bottom 7%: L= 6.04 cm

Top 7%: L= 6.22 cm

Step-by-step explanation:

Mean length of the population (μ) = 6.13 cm

Standard deviation (σ) = 0.06 cm

The z-score for any given length 'X' is:

[tex]z=\frac{X-\mu}{\sigma}[/tex]

What we want to know is the length at the 7-th percentile and at the 93-rd percentile.

According to a z-score table, the 7-th percentile has a correspondent z-score of -1.476 and the 93-rd percentile has a z-score of 1.476. Therefore, the bottom 7% and top 7% are separated by the following lengths:

[tex]z(X_B)=\frac{X_B-\mu}{\sigma}\\-1.476=\frac{X_B-6.13}{0.06}\\X_B = 6.04\\z(X_T)=\frac{X_T-\mu}{\sigma}\\1.476=\frac{X_T-6.13}{0.06}\\X_T = 6.22[/tex]

Bottom 7%: L= 6.04 cm.

Top 7%: L= 6.22 cm.