Answer:
a) The estimated percentile for a student who scores 425 on Writing is the 30.5th percentile.
b) The approximate score for a student who is at the 87th percentile for Writing is 613.5.
Step-by-step explanation:
Problems of normally distributed distributions are solved using the z-score formula.
In a set with mean
and standard deviation
, the zscore of a measure X is given by:
![Z = \frac{X - \mu}{\sigma}](https://tex.z-dn.net/?f=Z%20%3D%20%5Cfrac%7BX%20-%20%5Cmu%7D%7B%5Csigma%7D)
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 question, we have that:
![\mu = 484, \sigma = 115](https://tex.z-dn.net/?f=%5Cmu%20%3D%20484%2C%20%5Csigma%20%3D%20115)
a. What is the estimated percentile for a student who scores 425 on Writing?
This is the pvalue of Z when X = 425. So
![Z = \frac{X - \mu}{\sigma}](https://tex.z-dn.net/?f=Z%20%3D%20%5Cfrac%7BX%20-%20%5Cmu%7D%7B%5Csigma%7D)
![Z = \frac{425 - 484}{115}](https://tex.z-dn.net/?f=Z%20%3D%20%5Cfrac%7B425%20-%20484%7D%7B115%7D)
![Z = -0.51](https://tex.z-dn.net/?f=Z%20%3D%20-0.51)
has a pvalue of 0.3050.
The estimated percentile for a student who scores 425 on Writing is the 30.5th percentile.
b. What is the approximate score for a student who is at the 87th percentile for Writing?
We have to find X when Z has a pvalue of 0.87. So X for Z = 1.126.
![Z = \frac{X - \mu}{\sigma}](https://tex.z-dn.net/?f=Z%20%3D%20%5Cfrac%7BX%20-%20%5Cmu%7D%7B%5Csigma%7D)
![1.126 = \frac{X - 484}{115}](https://tex.z-dn.net/?f=1.126%20%3D%20%5Cfrac%7BX%20-%20484%7D%7B115%7D)
![X - 484 = 1.126*115](https://tex.z-dn.net/?f=X%20-%20484%20%3D%201.126%2A115)
![X = 613.5](https://tex.z-dn.net/?f=X%20%3D%20613.5)
The approximate score for a student who is at the 87th percentile for Writing is 613.5.