Answer:
Top 10%: 5.35 millimeters
Bottom 10%: 5.17 millimeters
Step-by-step explanation:
Problems of normally distributed samples can be solved using the z-score formula.
In a set with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the zscore 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 pvalue, we get the probability that the value of the measure is greater than X.
In this problem, we have that:
[tex]\mu = 5.26, \sigma = 0.07[/tex]
Top 10%
X when Z has a pvalue of 1-0.1 = 0.9. So X when Z = 1.28.
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]1.28 = \frac{X - 5.26}{0.07}[/tex]
[tex]X - 5.26 = 1.28*0.07[/tex]
[tex]X = 5.35[/tex]
Bottom 10%
X when Z has a pvalue of 0.1. So X when Z = -1.28
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]-1.28 = \frac{X - 5.26}{0.07}[/tex]
[tex]X - 5.26 = -1.28*0.07[/tex]
[tex]X = 5.17[/tex]
