Respuesta :
Answer:
The cutoff value for the top 5% of all the grades is 96.385.
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 = 75, \sigma = 13[/tex]
What is the approximate cutoff value for the top 5% of all the grades?
This is the value of X when Z has a pvalue of 0.95. So it is X when Z = 1.645.
So
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]1.645 = \frac{X - 75}{13}[/tex]
[tex]X - 75 = 13*1.645[/tex]
[tex]X = 96.385[/tex]
The cutoff value for the top 5% of all the grades is 96.385.
