The table shows the results for the years 2009 and 2011 of a citywide

Official Explanation

Statement 1: Seven is the straight average between the three average scores (6.4, 7.4, and 7.2) given by students, parents, and teachers, respectively. However, a different number of students, parents, and teachers responded, so a weighted average should be considered instead. Without performing the exact calculations, you can tell that, for instance, the 63,200 students weigh more heavily on the average than the 14,700 teachers do, so they will pull the average down compared to the straight average. (The actual average turns out to be 6.8.)

Answer: No

Statement 2: In 2011, the city’s schools received three ratings below 7.0: by students in the category of Communication and the category of Engagement, and by teachers in the category of Communication. Thus, among these three ratings, the probability that one of them was by students is 67% (two out of three).

Answer: Yes

Statement 3: The same principle applies here as in the first question. The straight average of 7.5 (the students’ grade) and 8.2 (the teachers’ grade) is 7.85. However, the weighted average is 7.61, which is lower than the parents’ grade of 7.8.

Answer: No