To find Jim's score, which is at the 80th percentile, we will first need to convert the percentile into a z-score. A z-score is a statistical measurement that describes a value's relationship to the mean of a group of values. It is used to quantify how many standard deviations an element is from the mean.
Start by converting the 80th percentile into a z-score using the inverse of the cumulative distribution function. For the 80th percentile, the z-score is approximately 0.842.
Then, we have to calculate the score. To do this, we transform the z-score back into the original scale. The formula used to calculate this is the z-score multiplied by the standard deviation plus the mean.
So, taking our z-score of 0.842 and multiply it by the standard deviation (which is 5), then add the mean (which is 21), we get Jim's score.
Therefore, Jim's score, which is at the 80th percentile, is approximately 25.21 on the ACT.