An investment company advertised that last year its clients

An investment company advertised that last year its clients, on average, made a profit of 9%.

Assuming that average refers to the mean, which of the following claims must be true based on this information?

Note: More than one statement could be true. If none of the statements is true, mark the appropriate box.

Last year, the number of their clients who made a profit of 9% or less was equal to the number of their clients who made a profit of 9% or more.

Last year all of their clients made a profit of at least 9%.

This year some of their clients will make a profit of at least 9%.

Last year at least one of their clients made a profit of less than 14%.

None of the above statements is true.

Solution

Last year at least one of their clients made a profit of exactly 9%. [FALSE, the mean may not be even present in the data set.]

Last year, the number of their clients who made a profit of 9% or less was equal to the number of their clients who made a profit of 9% or more. [FALSE. If the distribution is not perfectly symmetric, this is already false.]

Last year all of their clients made a profit of at least 9%. [FALSE.]

This year some of their clients will make a profit of at least 9%. [FALSE. We don\'t know what\'s going to happen next year for sure.]

Last year at least one of their clients made a profit of less than 14%. [TRUE. The mean is 9%, so someone must have gottn a profit less than 14%.]   

Thus, only this last statement is true.