4. The SAT and ACT are standardized tests for college admission in the United St

Question

4. The SAT and ACT are standardized tests for college admission in the United States. Both tests include components that measure reading comprehension. Suppose that SAT critical reading scores are Normally distributed with a mean of 510 and standard deviation of 115. In contrast, ACT reading scores are Normally distributed with a mean of 20.5 and standard deviation of 5. a) Sam takes the SAT reading test and scores 660. Dave takes the ACT test and scores 28. Who had the superior score, Sam or Dave? (5 points) (Hint: Standardize Dave’s and Sam’s scores and compare the probabilities.) b) How high must a student score to be in the top 10% of the ACT scores? (2 points) c) How high must a student score to be in the top 5% of the SAT scores? (2 points)

Explanation / Answer

a)

Dave's standardized score=(X-mean);/std deviation =(28-20.5)/5=1.5

Sam standardized score=(X-mean);/std deviation =(660-510)/115=1.30

as Dave has higher standardized score ; therefore Dave   had the superior score

b)

for top 10% score ; critical z =1.28

hence corresponding score =mean +z*std deviation =20.5+1.28*5=26.9

c)

for top 10% score ; critical z =1.645

hence corresponding score =mean +z*std deviation =510+1.645*115=699.175

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