Respuesta :
Answer:
A.
SAT score = 1060
ACT score = 23.2
B.
ACT score = 36.3
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 = 999, \sigma = 199[/tex]
If a student gets an SAT score that is the 62-percentile, find the actual SAT score.
This is X when Z has a pvalue of 0.62. So X when Z = 0.305.
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]0.305 = \frac{X - 999}{199}[/tex]
[tex]X - 999 = 0.305*199[/tex]
[tex]X = 1059.7[/tex]
Rounding to the nearest whole number.
SAT score = 1060
ACT:
[tex]\mu = 21.6, \sigma = 5.2[/tex]
The equivalent score is X when Z = 0.305.
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]0.305 = \frac{X - 21.6}{5.2}[/tex]
[tex]X - 21.6 = 0.305*5.2[/tex]
[tex]X = 23.19[/tex]
So
ACT score = 23.2
B. If a student gets an SAT score of 1563, find the equivalent ACT score
Z-score for the SAT score.
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{1563 - 999}{199}[/tex]
[tex]Z = 2.83[/tex]
Equivalent ACT:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]2.83 = \frac{X - 21.6}{5.2}[/tex]
[tex]X - 21.6 = 2.83*5.2[/tex]
[tex]X = 36.3[/tex]
ACT score = 36.3
