Final answer:
To calculate the z-score for a given SAT score, subtract the mean from the score and then divide by the standard deviation. The z-score represents the number of standard deviations a score is above or below the mean. Computing z-scores allows for comparison of performance across different normal distributions.
Step-by-step explanation:
The Scholastic Assessment Test (SAT) is typically modeled as a normal distribution with a certain mean (μ) and standard deviation (σ). To calculate a z-score, which indicates how many standard deviations a particular SAT score is from the mean, use the formula z = (x - μ) / σ.
To answer the parts of the shared question:
- a. For a score of 720 on the SAT with mean 520 and standard deviation 115:
z = (720 - 520) / 115 ≈ 1.74. This means the score of 720 is approximately 1.74 standard deviations above the mean. - b. A math SAT score that is 1.5 standard deviations above the mean of 520 is calculated by 520 + (1.5 × 115) ≈ 692.5. This score is considered high since it exceeds the average performance.
- c. To compare the SAT and ACT scores, calculate each test taker's z-score. The higher z-score will indicate which person did relatively better compared to others taking the same test.
The application and interpretation of the normal distribution and z-scores are vital components of statistics commonly used in various academic testing situations.