Questions: A weight-loss clinic advertised that last month its patients, on average, lost 8 lb. Assuming that average refers to the mean, which of the following claims must be true based on this information? This month some of their patients will lose at least 8 lb. Last month at least one of their patients lost exactly 8 lb. Last month all of their patients lost 8 lb or more. Last month, the number of their clients who lost more than 8 lb was equal to the number of their clients who lost less than 8 lb. Last month some of their patients lost more than 12 lb. None of the above statements are true.

Solution Steps

To determine which claims must be true based on the information that the average weight loss last month was 8 lb, we need to understand the properties of the mean. The mean is the sum of all values divided by the number of values. This does not necessarily imply that any individual value is exactly the mean, nor does it imply symmetry around the mean.

1. "This month some of their patients will lose at least 8 lb." - This statement is about future events and cannot be determined from past data.
2. "Last month at least one of their patients lost exactly 8 lb." - This is not necessarily true; the mean can be 8 lb without any individual patient losing exactly 8 lb.
3. "Last month all of their patients lost 8 lb or more." - This is not necessarily true; the mean can be 8 lb with some patients losing less than 8 lb.
4. "Last month, the number of their clients who lost more than 8 lb was equal to the number of their clients who lost less than 8 lb." - This is not necessarily true; the mean does not imply symmetry.
5. "Last month some of their patients lost more than 12 lb." - This is not necessarily true; the mean does not provide information about the maximum value.
6. "None of the above statements are true." - This is the most likely correct answer based on the analysis above.

• Analyze each statement to determine if it must be true based on the given mean.
• Conclude that none of the statements must be true.

The problem states that the average weight loss last month was 8 lb. This means the mean weight loss, \(\mu\), is 8 lb. The mean is calculated as: \[ \mu = \frac{\sum_{i=1}^{n} x_i}{n} \] where \(x_i\) represents the weight loss of the \(i\)-th patient and \(n\) is the total number of patients.

1. This month some of their patients will lose at least 8 lb.

   • This statement is about future events and cannot be determined from past data. Therefore, it is not necessarily true.

2. Last month at least one of their patients lost exactly 8 lb.

   • The mean being 8 lb does not imply that any individual patient lost exactly 8 lb. Therefore, this statement is not necessarily true.

3. Last month all of their patients lost 8 lb or more.

   • The mean being 8 lb does not imply that all patients lost 8 lb or more. Some could have lost less. Therefore, this statement is not necessarily true.

4. Last month, the number of their clients who lost more than 8 lb was equal to the number of their clients who lost less than 8 lb.

   • The mean does not imply symmetry around the mean. Therefore, this statement is not necessarily true.

5. Last month some of their patients lost more than 12 lb.

   • The mean being 8 lb does not provide information about the maximum value. Therefore, this statement is not necessarily true.

6. None of the above statements are true.

   • Based on the analysis above, none of the previous statements must be true.

Final Answer