Final answer:
To determine specific percentiles and score ranges on the SAT, we utilize the mean and standard deviation within the context of a normal distribution. The 80th percentile gives the minimum score for the top 20%, while the 10th and 90th percentiles provide the bounds for the middle 80%.
Step-by-step explanation:
Understanding SAT scores and normal distribution
When analyzing SAT scores, it is essential to use the principles of normal distribution. Given that SAT scores follow a normal distribution with a known mean (μ) and standard deviation (σ), we can utilize these parameters to calculate specific percentile scores and ranges.
a) To find the minimum score necessary to be in the top 20% of the SAT distribution, we look for the 80th percentile, since the top 20% is equivalent to the bottom 80%. This involves finding the z-score that corresponds to the 80th percentile, and then using the mean and standard deviation to convert that z-score to an SAT score.
b) The range of values that defines the middle 80% of the distribution is found by identifying the 10th and 90th percentiles. These percentiles demarcate the lower and upper bounds of the middle 80% of SAT scores, encompassing the majority of test-takers.