Let us analyze each statement in light of the information given (that the mean SAT score of the college's freshman students last year was 1120):

Analysis of the Statements

1. Statement (a): Next year at least one of their freshman students will have a score of at least 1120 on the SAT.

   • This statement refers to the future, but the information about the mean score pertains to last year's students only. We cannot make any claims about next year's scores based on this data.
   • This statement is false.

2. Statement (b): Last year some of their freshman students had a score of exactly 1120 on the SAT.

   • The mean does not guarantee that any individual student scored exactly 1120. It is possible for no student to have scored this exact value if their scores were distributed around the mean.
   • This statement is false.

3. Statement (c): Last year at least one of their freshman students had a score of less than 1290 on the SAT.

   • The mean is 1120, which is significantly below 1290. It is impossible for all students to have scores above 1290, as this would make the mean score much higher than 1120. Thus, at least one student must have had a score below 1290.
   • This statement is true.

4. Statement (d): Last year more than half of their freshman students had a score of at least 1120 on the SAT.

   • The mean does not guarantee that more than half of the scores are greater than or equal to it. This would require knowledge of the median, which we do not have. It is possible for most scores to be below the mean while a few high scores pull the average up.
   • This statement is false.

5. Statement (e): Last year all of their freshman students had a score of at least 1120 on the SAT.

   • If all students scored 1120 or higher, the mean would equal or exceed the lowest score, i.e., 1120. However, the mean could still be 1120 with some scores below it, so this statement is not necessarily true.
   • This statement is false.

6. Statement (f): None of the above statements are true.

   • Since Statement (c) is true, this option is incorrect.

Correct Answer:

The only true statement is (c): "Last year at least one of their freshman students had a score of less than 1290 on the SAT."

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Would you like further clarification or a deeper explanation of any part of this problem? Here are 5 related questions for you to explore:

1. What is the difference between the mean, median, and mode in statistics?
2. How does the distribution of data affect the interpretation of the mean?
3. Could the mean score of 1120 indicate that some students scored exceptionally low or high? Why or why not?
4. If the median SAT score was provided, how could it change the interpretation of these statements?
5. Why is it impossible for all students to score higher than the mean value?

Tip: The mean alone does not provide the complete picture of a dataset; consider additional measures like the median or standard deviation to better understand the distribution.