Final answer:
To answer the student's question, about 68% of SAT math scores will be between 411 and 639, about 32% will be less than 411 or greater than 639, and around 2.5% will be greater than 753. These calculations are based on the properties of a normal distribution with given mean and standard deviation.
Step-by-step explanation:
The student's question pertains to the application of normal distribution and standard deviation to assess SAT math scores. This involves calculating percentages of scores that fall within certain ranges, as well as understanding z-scores and comparisons across different standardized tests.
- To determine what percentage of standardized test scores is between 411 and 639, we would use the properties of normal distribution. In a normal distribution, approximately 68% of all data falls within one standard deviation of the mean. Since 411 is one standard deviation below the mean (525 - 114) and 639 is one standard deviation above the mean (525 + 114), approximately 68% of test scores fall between these two values.
- To find what percentage of standardized test scores is less than 411 or greater than 639, we look at the remaining percentages outside of one standard deviation from the mean. Since 68% fall within one standard deviation, the remaining 32% is split between the lower and upper tails, so 16% of scores are less than 411 and 16% are greater than 639.
- To calculate the percentage of standardized test scores that is greater than 753, we find that 753 is two standard deviations above the mean (525 + 2(114)). In a normal distribution, 95% of the scores fall within two standard deviations of the mean, which leaves 5% in the tails. Since the scores are split between the lower and upper tails, only 2.5% of scores will be greater than 753.