Final answer:
To find a z-score, subtract the mean from the score and divide by the standard deviation. For comparative purposes, SAT and ACT scores must be converted to z-scores. The z-score allows us to understand a score's position relative to the average.
Step-by-step explanation:
The student question pertains to calculating probabilities and z-scores using the properties of a normal distribution. A z-score is found by subtracting the mean from the score in question and then dividing by the standard deviation (z = (x - μ) / σ). The probability related to a specific z-score can then be obtained using a z-table or statistical software.
For example, the z-score for an SAT score of 720 can be calculated using the formula, where 520 is the mean, and 115 is the standard deviation. This would be z = (720 - 520) / 115 ≈ 1.74. A z-score of 1.74 indicates that a score of 720 is 1.74 standard deviations above the mean. To calculate a math SAT score 1.5 standard deviations above the mean, we multiply the standard deviation by 1.5 and add it to the mean, which gives us 520 + 1.5(115) ≈ 692.5.
To compare scores from different distributions, like the SAT and ACT, we need to calculate z-scores for each and compare them. For instance, if a student scores 700 on the SAT and another scores 30 on the ACT, each score must first be transformed into a z-score using their respective mean and standard deviation, then they can be directly compared.