Answer:
Luis would need to have a SAT score of 574.
Step-by-step explanation:
Normal Probability Distribution:
Problems of normal distributions can be solved using the z-score formula.
In a set with mean
and standard deviation
, the z-score 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 p-value, we get the probability that the value of the measure is greater than X.
Nicole's z-score:
ACT scores have a mean of about 21 with a standard deviation of about 5, which means that ![\mu = 21, \sigma = 5](https://tex.z-dn.net/?f=%5Cmu%20%3D%2021%2C%20%5Csigma%20%3D%205)
Nicole gets a score of 24, which means that
. Her z-score is:
![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{24 - 21}{5}](https://tex.z-dn.net/?f=Z%20%3D%20%5Cfrac%7B24%20-%2021%7D%7B5%7D)
![Z = 0.6](https://tex.z-dn.net/?f=Z%20%3D%200.6)
What score would Luis have to have on the SAT to have the same standardized score(z-score) as Nicole's standardized score on the ACT?
Luis would have to get a score with a z-score of 0.6, that is, X when Z = 0.6.
SAT scores have a mean of about 508 with a standard deviation of about 110, which means that
.
The score is:
![Z = \frac{X - \mu}{\sigma}](https://tex.z-dn.net/?f=Z%20%3D%20%5Cfrac%7BX%20-%20%5Cmu%7D%7B%5Csigma%7D)
![0.6 = \frac{X - 508}{110}](https://tex.z-dn.net/?f=0.6%20%3D%20%5Cfrac%7BX%20-%20508%7D%7B110%7D)
![X - 508 = 0.6*110](https://tex.z-dn.net/?f=X%20-%20508%20%3D%200.6%2A110)
![X = 574](https://tex.z-dn.net/?f=X%20%3D%20574)
Luis would need to have a SAT score of 574.