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Answer:
The diameter that separates the bottom 9% is 5.02 millimeters.
The diameter that separates the top 9% is 5.2 millimeters.
Step-by-step explanation:
Normal Probability Distribution:
Problems of normal distributions can be solved using the z-score formula.
In a set with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the z-score of a measure X is given by:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
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 5.11 millimeters and a standard deviation of 0.07 millimeters.
This means that [tex]\mu = 5.11, \sigma = 0.07[/tex]
Bottom 9%:
The 9th percentile, which is X when Z has a pvalue of 0.09. So X when Z = -1.34.
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]-1.34 = \frac{X - 5.11}{0.07}[/tex]
[tex]X - 5.11 = -1.34*0.07[/tex]
[tex]X = 5.02[/tex]
The diameter that separates the bottom 9% is 5.02 millimeters.
Top 9%:
The 100 - 9 = 91th percentile, which is X when Z has a pvalue of 0.91. So X when Z = 1.34.
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]1.34 = \frac{X - 5.11}{0.07}[/tex]
[tex]X - 5.11 = 1.34*0.07[/tex]
[tex]X = 5.2[/tex]
The diameter that separates the top 9% is 5.2 millimeters.
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