Respuesta :
Answer:
The value that separates the top 5% is 5.86 millimeters.
The value that separates the bottom 5% is 5.62 millimeters.
Step-by-step explanation:
Problems of normally distributed samples are 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:
Top 5%
value of X when Z has a pvalue of 1-0.05 = 0.95. So X when Z = 1.645.
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]1.645 = \frac{X - 5.74}{0.07}[/tex]
[tex]X - 5.74 = 0.07*1.645[/tex]
[tex]X = 5.86[/tex]
The value that separates the top 5% is 5.86 millimeters.
Bottom 5%
value of X when Z has a pvalue of 0.05. So X when Z = -1.645
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]-1.645 = \frac{X - 5.74}{0.07}[/tex]
[tex]X - 5.74 = 0.07*(-1.645)[/tex]
[tex]X = 5.62[/tex]
The value that separates the bottom 5% is 5.62 millimeters.
