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In 2017 the SAT had a mean score of 1060 with standard deviation of 195 (max score of 1600) while the ACT had a mean of 20.9 with standard deviation of 5.6 (max composite score of 36). Reed and Maria apply to the same college. Maria took the SAT and received a score of 1270 while Reed took the ACT and received a score of 27. Which student performed better in relationship to their peers

Sagot :

Answer:

Due to the higher z-score, Reed performed better in relationship to their peers

Step-by-step explanation:

Z-score:

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.

In this question:

Whoever had the higher z-score performed better in relation to their peers.

Maria:

Took the SAT, grade 1270, so [tex]X = 1270[/tex]

Mean score of 1060 with standard deviation of 195 (max score of 1600). This means that [tex]\mu = 1060, \sigma = 295[/tex]. So

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

[tex]Z = \frac{1270 - 1060}{295}[/tex]

[tex]Z = 0.71[/tex]

Reed:

Took the ACT, score of 27, so [tex]X = 27[/tex]

Mean of 20.9 with standard deviation of 5.6, which means that [tex]\mu = 20.9, \sigma = 5.6[/tex]. So

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

[tex]Z = \frac{27 - 20.9}{5.6}[/tex]

[tex]Z = 1.09[/tex]

Due to the higher z-score, Reed performed better in relationship to their peers

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