Question

an investment company advertised that last year its clients, on average, made a profit of 7%. 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 all of their clients made a profit of at least 7%.
last year, the number of their clients who made a profit of less than 7% was equal to the number of their clients who made a profit of more than 7%.
this year some of their clients will make a profit of at least 7%.
last year at least one of their clients made a profit of exactly 7%.
last year some of their clients made a profit of more than 3%.
none of the above statements are true.

Explanation:

Analyze the definition of the mean

Using the Sample Mean knowledge point
The arithmetic mean is the sum of all values divided by the total number of values:

$$ \mu = \frac{\sum_{i=1}^{n} x_i}{n} = 7\% $$

This represents the central balance point of the data, but does not dictate individual values.

Evaluate statements 1 and 2

Using the Sample Mean knowledge point

• Statement 1: "Last year all of their clients made a profit of at least 7%."

This is false. Some clients could make less than 7% (or even lose money) as long as others make more than 7% to balance the average.

• Statement 2: "Last year, the number of their clients who made a profit of less than 7% was equal to the number of their clients who made a profit of more than 7%."

This is false. Equal counts above and below describes the median, not the mean. A skewed distribution can have a mean of 7% with unequal counts on either side.

Evaluate statements 3 and 4

Using the Sample Mean knowledge point

• Statement 3: "This year some of their clients will make a profit of at least 7%."

This is false. The advertisement only provides historical data about "last year" and cannot guarantee future performance.

• Statement 4: "Last year at least one of their clients made a profit of exactly 7%."

This is false. It is possible for no individual client to have a profit of exactly 7% (e.g., two clients with profits of 6% and 8% have a mean of 7%).

Evaluate statement 5

Using the Sample Mean knowledge point

• Statement 5: "Last year some of their clients made a profit of more than 3%."

This must be true. If all clients made a profit of 3% or less, the maximum possible mean would be 3%. Since the mean is 7%, at least one client must have made a profit strictly greater than 3% (and indeed, greater than or equal to 7%).

Evaluate statement 6

Using the Sample Mean knowledge point

• Statement 6: "None of the above statements are true."

Since Statement 5 is true, this statement is false.

Answer:

• Last year all of their clients made a profit of at least 7%.
• Last year, the number of their clients who made a profit of less than 7% was equal to the number of their clients who made a profit of more than 7%.
• This year some of their clients will make a profit of at least 7%.
• Last year at least one of their clients made a profit of exactly 7%.
• Last year some of their clients made a profit of more than 3%. (Correct answer)
• None of the above statements are true.