Question

an investment company advertised that last year its clients, on average, made a profit of 8%. assuming that average refers to the mean, which of the claims must be true based on this information? note: more than one statement could be true. if none of the statements are true, mark the appropriate box. last year some of their clients made a profit of less than 14%. last year at least one of their clients made a profit of exactly 8%. last year all of their clients made a profit of 8% or more. last year more than half of their clients made a profit of at least 8%. two years ago some of their clients made a profit of at least 8%. none of the above statements are true.

Explanation:

Step1: Recall the concept of mean

The mean (average) of a set of data is the sum of all values divided by the number of values. Just because the mean profit of clients is 8% doesn't mean any specific client made exactly 8%. Some clients could have made more and some could have made less.

Step2: Analyze each statement

• For "Last year some of their clients made a profit of less than 14%": Since the mean is 8%, it is very likely (in fact, almost certain in non - degenerate cases) that some clients made less than 14%. For example, if some clients made very high profits, others must have made lower profits to bring the average down to 8%.
• For "Last year at least one of their clients made a profit of exactly 8%": The mean being 8% does not imply that any client made exactly 8%. It is possible that all clients made profits other than 8% but when summed and divided by the number of clients, the result is 8%.
• For "Last year all of their clients made a profit of 8% or more": If all clients made 8% or more, the mean would be 8% or more. But the mean is 8%, so it is not possible that all clients made 8% or more. There must be some clients who made less than 8% to balance out any clients who made more than 8%.
• For "Last year more than half of their clients made a profit of at least 8%": There is no information to support this. The mean of 8% can be achieved in many ways, such as having a few clients with very high profits and many clients with low profits.
• For "Two years ago some of their clients made a profit of at least 8%": The information given is about last year, so we cannot make any conclusions about two years ago.

Answer:

Last year some of their clients made a profit of less than 14%.