When the Scholastic Aptitude Test (SAT) - now known as the Scholastic Assessment Test - was first developed, the idea was that each section of the test would have a mean of 500 and a standard deviation of 100. While the SAT is continually adjusted and readjusted to promote such a distribution, let's assume for this exercise that math scores on the SAT have a mean of 500 and a standard deviation of 100.
- Suppose that a student receives a 465 on the math section of the SAT. What is the z-score that corresponds to this raw score?
- Interpret your answer from Question #1. What does this z-score represent in terms of the explanations provided in Chapter 6?
- What percentage of students who took the SAT scored below 465? What percentage scored above 465?
- What percentage of students who took the SAT scored between a 465 and 500?
- On the basis of your answers to the above questions, what percentile does a score of 465 correspond to?
- Suppose that a second student scored a 595 on the math portion of the SAT. Calculate the z-score associated with this raw score.
- What percentage of students scored above 595 on the math portion of the SAT?
- What is the percentile rank associated with a score of 595?
- What percentage of students who took the SAT posted scores between the two students discussed in this exercise?
- Assume for the moment that the first student who scored a 465 retook the SAT and posted a z-score of .35, representing a marked improvement over his or her previous score. What percentage of students who took the SAT posted scores between a z-score of .35 and a raw score of 595?
- Finally, suppose three siblings each take the SAT. The first two siblings post scores below the mean of 415 and 425 respectively. The third sibling however posts a score of 610. Calculate the proportion of students who took the SAT who either scored between 415 and 425 or above 610.
- What do we need to assume about the distribution of the SAT scores to create z-scores?
- Assuming the distribution of the SAT scores is normal, what can you say about the median, mean, and mode?