Final answer:
The z-score for an SAT score of 720 is about 1.74, indicating it is 1.74 standard deviations above the mean. A math SAT score that is 1.5 standard deviations above the mean would be approximately 692.5. Comparing the SAT and ACT math scores through z-scores indicates that a score of 30 on the ACT is slightly higher relative to the mean than a score of 700 on the SAT, with z-scores of 1.70 and 1.59 respectively.
Step-by-step explanation:
Calculating SAT Scores and Understanding Distribution
To calculate the z-score for an SAT score of 720 when the mean (µ) is 520 and the standard deviation (o) is 115, we use the formula:
z = (X - µ)/o
Where X is the score of interest (720 in this case). Plugging in the numbers:
z = (720 - 520) / 115 ≈ 1.74
This z-score of approximately 1.74 indicates that a score of 720 is 1.74 standard deviations above the mean SAT mathematics score.
To find what math SAT score is 1.5 standard deviations above the mean, we calculate:
Score = µ + (z * o)
Score = 520 + (1.5 * 115) ≈ 692.5
This score is relatively high as it is above the average and suggests that the student performed quite well.
When comparing SAT and ACT math test scores to determine which person did better relative to their respective test, we convert each score into its corresponding z-score.
For the SAT math test score of 700:
z = (700 - 514) / 117 ≈ 1.59
For the ACT score of 30:
z = (30 - 21) / 5.3 ≈ 1.70
The person with the ACT score of 30 has a slightly higher z-score, indicating they did slightly better in terms of standard deviations above the mean, but this only indicates performance relative to others who took the same test, not a direct comparison between the two tests.