Respuesta :
Answer:
a) 38th percentile.
b) 631
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 = 484, \sigma = 115[/tex]
(a) What is the estimated percentile for a student who scores 450 on Writing? Round your answer to the nearest integer.
This is the pvalue of Z when X = 450. So
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{450 - 484}{115}[/tex]
[tex]Z = -0.3[/tex]
[tex]Z = -0.3[/tex] has a pvalue of 0.3821.
So the estimated percentile for a student who scores 450 on Writing is the 38th percentile.
(b) What is the approximate score for a student who is at the 90th percentile for Writing? Round your answer to the nearest integer.
Value of X when Z has a pvalue of 0.9. So X when Z = 1.28.
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]1.28 = \frac{X - 484}{115}[/tex]
[tex]X - 484 = 1.28*115[/tex]
[tex]X = 631.2[/tex]
Rounded to the nearest integer, 631
