Ann from Example 2.3.1 earned a score of 1800 on her SAT with a corresponding . She would like to know what quantile she falls in among all SAT test-takers.
Answer. Ann’s quantile is the percentage of people who earned a lower SAT score than Ann. We shade the area representing those individuals in Figure LABEL:satBelow1800. The total area under the normal curve is always equal to 1, and the proportion of people who scored below Ann on the SAT is equal to the area shaded in Figure LABEL:satBelow1800: . In other words, Ann is in the quantile of SAT takers. We can use the normal model to find quantiles. A normal probability table, which lists Z scores and corresponding quantiles, can be used to identify a quantile based on the Z score (and vice versa). This is probably how you identified probabilities in A level. For this course we will use statistical software although in the exam a range of probabilities will be given (as in the Appendix) and you will have to choose which one(s) to use.
In R we use the pnorm function to calculate Normal probabilities. For example 2.3.4 this would be
R> pnorm(1800,mean=1500,sd=300)
We can also find the Z score associated with a quantile. For example, to identify Z for the quantile in R we use the qnorm function, so this would be
R> qnorm(0.8)
Determine the proportion of SAT test takers who scored better than Ann on the SAT.
Answer. If 84% had lower scores than Ann, the number of people who had better scores must be 16%. (Generally ties are ignored when the normal model, or any other continuous distribution, is used.)