To determine which statement must be true given that the average of a group of numbers is -0.75, we will carefully consider what an average of -0.75 implies.
1. Understanding the Average:
- The average (or mean) of a set of numbers is calculated by dividing the sum of all the numbers by the count of the numbers.
- If the average is -0.75, it means:
[tex]\[
\text{Average} = \frac{\text{Sum of all numbers}}{\text{Count of numbers}} = -0.75
\][/tex]
2. Implication of a Negative Average:
- For the average to be negative, the sum of all the numbers in the set must be a negative number. This is because any positive sum divided by a positive count would result in a positive average, which contradicts the given average of -0.75.
3. Evaluating the Statements:
- A. There must be 4 numbers in the group.
- This statement suggests a specific count of numbers, but the count of numbers is not inherently determined by the given average. Any count could achieve an average of -0.75, provided the sum is correct.
- B. All of the numbers in the group must be negative.
- This statement suggests that every individual number must be negative. However, this is not necessarily true. For example, a mixture of positive and negative numbers can still yield a negative average if the negative numbers outweigh the positive ones sufficiently.
- C. The sum of the numbers in the group must be negative.
- As reasoned earlier, for the average to be -0.75, the sum divided by the count must lead to a negative value. Hence, the sum itself must be negative to obtain a negative average.
- D. The sum of the numbers in the group could be positive.
- This is directly contradictory because a positive sum divided by a positive count would result in a positive average, not a negative one as given.
4. Conclusion:
- Given the negative average, the only statement that must be true is:
C. The sum of the numbers in the group must be negative.
This ensures that the average of the group of numbers results in the correct negative value as specified.
