Respuesta :
Answer:
The actual SAT score is 1260.6.
The equivalent ACT score for this student(77th percentile) is 24.7.
The equivalent ACT score to a SAT score of 1544 is 30.26.
Step-by-step explanation:
When the distribution is normal, we use 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.
SAT:
[tex]\mu = 1117, \sigma = 194[/tex]
ACT:
[tex]\mu = 21.9, \sigma = 3.8[/tex]
If a student gets an SAT score that is the 77-percentile, find the actual SAT score.
This is X when Z has a pvalue of 0.77. So X when Z = 0.74.
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]0.74 = \frac{X - 1117}{194}[/tex]
[tex]X - 1117 = 0.74*194[/tex]
[tex]X = 1260.6[/tex]
The actual SAT score is 1260.6.
What would be the equivalent ACT score for this student?
X when Z = 0.74, using [tex]\mu = 21.9, \sigma = 3.8[/tex]
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]0.74 = \frac{X - 21.9}{3.8}[/tex]
[tex]X - 21.9 = 0.74*3.8[/tex]
[tex]X = 24.7[/tex]
The equivalent ACT score for this student(77th percentile) is 24.7.
If a student gets an SAT score of 1544, find the equivalent ACT score.
First we find the z-score of the SAT, then we find the equivalent ACT score.
Z-score of SAT:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{1544 - 1117}{194}[/tex]
[tex]Z = 2.2[/tex]
Equivalent ACT:
X when Z = 2.2. So
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]2.2 = \frac{X - 21.9}{3.8}[/tex]
[tex]X - 21.9 = 2.2*3.8[/tex]
[tex]X = 30.26[/tex]
The equivalent ACT score to a SAT score of 1544 is 30.26.
