Answer to Question #90734 in Statistics and Probability for Ray mcDonnell

Question

John also measured the heights of the teachers in his school.
John says: “Most of the teachers in my school have a height that is greater than 175 cm.”
Put a tick in the correct box to show which of the following statements must be true, based on what John says. Tick () one box only. Give a reason for your answer.
The mean height of the teachers is greater than 175 cm
The median height of the teachers is greater than 175 cm
The modal height of the teachers (the mode) is greater than 175 cm

Answer 1

1)Counterexample: If in the school only three teacher and their heights are 150 cm., 180 cm. and 180 cm.,

then most of them have a height greater than 175 cm

but mean height is (150 + 180 + 180)/3 = 170 < 175.

The mean height of the teachers can be less than 175 cm. even if most of the teachers have a height that is greater than 175 cm.

2)Proof by contradiction, that median height of the teachers is greater than 175 cm.

Suppose median height of the teachers isn't greater than 175 cm , then not fewer than half of the teachers have height not greater than 175 cm., hence no more than half of the teachers have a height greater than 175 cm. contradiction, therefore assumption ' median height of the teachers not greater than 175 cm' is false, and median height of the teachers is greater than 175 cm.

If most of the teachers have a height that is greater than 175 cm., than median height of the teachers is greater than 175 cm.

3)Counterexample:  If in the school teachers have heights 170 cm., 170 cm., 176 cm., 177 cm., 178 cm., then most of them have height greater than 175 cm., but the most frequent height is 170 cm. and mode height is 170 cm.< 175 cm.

The modal height of the teachers can be less than 175 cm. even if most of the teachers have a height that is greater than 175 cm.

Answer: 1)False because the mean height of the teachers can be less than 175 cm., 2)True because the median height of the teachers is greater than 175 cm., 3)False because the modal height of the teachers can be less than 175 cm.