Question

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?
note: more than one statement could be true. if none of the statements is true, mark the appropriate box.

• last month more than half of their patients lost at least 8 lb.
• last month some of their patients lost exactly 8 lb.
• this month at least one of their patients will lose at least 8 lb.
• last month all of their patients lost more than 4 lb.
• last month some of their patients lost at least 8 lb.
• none of the above statements are true.

Explanation:

Analyze the given information

Using the Sample Mean knowledge point
The clinic states that last month, the mean weight loss of its patients was \(8\text{ lb}\). Let \(n\) be the number of patients, and \(x_i\) be the weight loss of patient \(i\).

$$ \bar{x} = \frac{1}{n} \sum_{i=1}^{n} x_i = 8 $$

Evaluate Statement 1 and Statement 2

Using the Mean Properties knowledge point

• Statement 1: "Last month more than half of their patients lost at least \(8\text{ lb}\)."

This describes the median, not the mean. A skewed distribution (e.g., one patient losing \(80\text{ lb}\) and nine patients losing \(0\text{ lb}\)) yields a mean of \(8\text{ lb}\), but only \(10\%\) of patients lost at least \(8\text{ lb}\). Thus, this statement is not necessarily true.

• Statement 2: "Last month some of their patients lost exactly \(8\text{ lb}\)."

The mean can be \(8\text{ lb}\) without any individual value being exactly \(8\text{ lb}\) (e.g., two patients losing \(6\text{ lb}\) and \(10\text{ lb}\)). Thus, this statement is not necessarily true.

Evaluate Statement 3 and Statement 4

Using the Mean Properties knowledge point

• Statement 3: "This month at least one of their patients will lose at least \(8\text{ lb}\)."

Past performance (last month) does not guarantee future outcomes (this month). Thus, this statement is not necessarily true.

• Statement 4: "Last month all of their patients lost more than \(4\text{ lb}\)."

Some patients could have lost less than \(4\text{ lb}\), or even gained weight, as long as others lost enough to keep the average at \(8\text{ lb}\). Thus, this statement is not necessarily true.

Evaluate Statement 5

Using the Mean Properties and Statistical Interpretation knowledge points

• Statement 5: "Last month some of their patients lost at least \(8\text{ lb}\)."

If the average weight loss is \(8\text{ lb}\), it is mathematically impossible for every single patient to have lost strictly less than \(8\text{ lb}\). If \(x_i < 8\) for all \(i\), then:

$$ \sum_{i=1}^{n} x_i < 8n \implies \bar{x} < 8 $$

This contradicts the given mean of \(8\text{ lb}\). Therefore, at least one patient (some patients) must have lost at least \(8\text{ lb}\). This statement must be true.

Answer:

• Last month more than half of their patients lost at least 8 lb.
• Last month some of their patients lost exactly 8 lb.
• This month at least one of their patients will lose at least 8 lb.
• Last month all of their patients lost more than 4 lb.
• Last month some of their patients lost at least 8 lb. (Correct answer)
• None of the above statements are true.