High school students are faced with taking the ACT or the SAT tests to measure t

Question

High school students are faced with taking the ACT or the SAT tests to measure their readiness to start college. Quantitative SAT scores have a mean of 500 and a standard deviation of 100 while quantitative ACT scores have a mean of 21 and a standard deviation of 5. A student takes the ACT and receives a quantitative score of 2% while another takes the SAT and receives a quantitative score of 750. Which student has scored higher relatively speaking? What is the ACT quantitative score that separates the top 25%? A student receives a SAT quantitative score of 850. What percentile is the student in?

Explanation / Answer

Question 3

We are given

For SAT score, mean = 500, SD = 100

For ACT score, mean = 21, SD = 5

Part a

Here, we have to compare the z-scores for both students. The Z-score formula is given as below:

Z = (X – mean) / SD

Z score for student who takes the ACT is given as below:

Z = (28 – 21) / 5

Z = 1.4

Z score for student who takes the SAT is given as below:

Z = (750 – 500) / 100

Z = 2.5

Z-score for student who takes the SAT is more than the Z-score for student who takes ACT.

The student takes the SAT; score higher as compare to the student with ACT.

Part b

For ACT score, mean = 21, SD = 5

Z = (X – mean) / SD

X = Mean + Z*SD

Z for top 25% or lower 75% = 0.67449

X = 21 + 0.67449*5 = 24.37245

Required ACT quantitative score = 24.37245

Part c

Here, we have to find P(X<850)

For SAT score, mean = 500, SD = 100

Z = (X – mean) / SD

Z = (850 – 500) / 100

Z = 350/100 = 3.5

P(X<850) = P(Z<3.5) = 0.9998

Required answer: 99th percentile

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