Regional Academic Standards: A Comparative Statistical Study

Cumulative frequency graph for Mathematics and Science examinations

The cumulative frequency graph shows the distribution of marks for \(400\) students in a Mathematics and Science examination.

1. 1. Find the median mark for the Mathematics examination.
   2. Find the interquartile range for the Science examination.
   3. If the distinctions for Mathematics were allocated to the top \(25\%\) of the students, find the mark to achieve a distinction.
2. One student is chosen at random. You may assume that the marks gained in the two examinations are independent. Expressing your answer as a fraction in its lowest terms, calculate the probability that the student gains
   1. less than \(20\) marks for both examinations,
   2. more than \(60\) marks on one examination, but not on the other.