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There are two major tests of readiness for college: the ACT and the SAT. ACT scores are reported on a scale from 1 to 36. The distribution of ACT scores for more than 1 million students in a recent high school graduating class was roughly normal with mean [tex]\mu = 15.6[/tex] and standard deviation [tex]\sigma = 3.9[/tex]. SAT scores are reported on a scale from 400 to 1600. The SAT scores for 1.4 million students in the same graduating class were roughly normal with mean [tex]\mu = 999[/tex] and standard deviation [tex]\sigma = 190[/tex].

Jose scores 927 on the SAT. Assuming that both tests measure the same thing, what score on the ACT is equivalent to Jose's SAT score?

Answer :

Answer:

An ACT score of 14.118 is equivalent to Jose's SAT score.

Step-by-step explanation:

The ACT equiavalent to Jose's SAT score is going to be ACT grade with the same z-score as Jose's SAT grade.

The z-score formula is given by:

[tex]Z = \frac{X - \mu}{\sigma}[/tex]

In which: X is Jose's grade, [tex]\mu[/tex] is the mean value of all the grades and [tex]\sigma[/tex] is the standard deviation of the grades.

About the SAT test, we have the following information:

Mean of 999. So [tex]/mu = 999[/tex]

Standard deviation of 190. So [tex]\sigma = 190[/tex]

Jose scored 927. So [tex]X = 927[/tex]

The z-score of Jose's SAT grade is:

[tex]Z = \frac{X - \mu}{\sigma} = \frac{927 - 999}{190} = -0.38[/tex]

About the ACT test, we have the following information:

Mean of 15.6, so [tex]\mu = 15.6[/tex]

Standard deviation of 3.9, so [tex]\sigma = 3.9[/tex]

We want to find the value o X when [tex]Z = -0.38[/tex]. So:

[tex]Z = \frac{X - \mu}{\sigma}[/tex]

[tex]-0.38 = \frac{X - 15.6}{3.9}[/tex]

[tex]X - 15.6 = -1.482[/tex]

[tex]X = 14.118[/tex]

An ACT score of 14.118 is equivalent to Jose's SAT score.

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