Answer:
P6 = 88.3715
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:
[tex]\mu = 195.2, \sigma = 68.7[/tex]
Find P6, which is the score separating the bottom 6% from the top 94%.
This is the value of X when Z has a pvalue of 0.06. So it is X when Z = -1.555.
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]-1.555 = \frac{X - 195.2}{68.7}[/tex]
[tex]X - 195.2 = -1.555*68.7[/tex]
[tex]X = 88.3715[/tex]
P6 = 88.3715
