"In the year 2005 National exam, 57.4% of the students who sat for the mathematics paper obtained grade A. A private teacher claims that his record is superior, because 65 out of his 100 students who took the same exam obtained grade A.

1. Is his claim justified at the 2% level of significance?

2. What is the smallest level of significance at which his claim would be justified?"

I feel that there is not enough information to carry out such a test, for the simple reason that the population consists of groups of students who took private tuition and obtained grade A (call this group X), as well as those who did not take private tuition and obtained A (call this group Y). The sample proportion of 0.65 should be compared with the proportion of grade A within X but not with the population proportion 0.574. But information on the proportion within X is missing. Therfore no meaningful comparison could be made.