Most college-bound students take either the SAT(Scholastic Assessment Test) or the ACT (which originally stood for American college testing). Scores on both the ACT and the SAT are approximately normally distributed. ACT scores have a mean of about 21 with a standard deviation of about 5.SAT scores have a mean of about 508 with a standard deviation of about 110. Nicole takes the ACT and gets a score of 24. Luis takes the SAT. 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?

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 [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the z-score 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 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 [tex]\mu = 21, \sigma = 5[/tex]

Nicole gets a score of 24, which means that [tex]X = 24[/tex]. Her z-score is:

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

[tex]Z = \frac{24 - 21}{5}[/tex]

[tex]Z = 0.6[/tex]

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 [tex]\mu = 508, \sigma = 110[/tex].

The score is:

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

[tex]0.6 = \frac{X - 508}{110}[/tex]

[tex]X - 508 = 0.6*110[/tex]

[tex]X = 574[/tex]

Luis would need to have a SAT score of 574.

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