Introduction

A common problem in statistics and mathematics is determining mean scores under various conditions. For instance, a class of 52 students has a mean score of 64 on a statistics examination. If we know the mean score of a subset of 34 students is between 60 and 67, we can calculate the lowest possible mean score for the remaining 18 students. This problem not only strengthens our understanding of mean scores but also enhances our problem-solving skills in statistical analysis.

Problem Statement

A class of 52 students has a mean score of 64 on a statistics examination. The mean score of a subset of 34 students is between 60 and 67. We need to find the lowest possible mean score for the remaining 18 students.

Step-by-Step Solution

Step 1: Calculating the Total Score for All Students

The total score for all 52 students can be calculated as follows:

Total score for all students Mean score times; Number of students

Total score for 52 students 64 times; 52 3328

Step 2: Determining the Total Score for the 34 Students

Let the mean score of the 34 students be x. Since the mean score is between 60 and 67, we can express the total score for these 34 students as:

Total score for 34 students x times; 34

We will maximize x to find the lowest possible mean score for the remaining 18 students. The maximum value for x is just below 67 (67 - ε), where ε is a very small positive number.

Step 3: Expressing the Total Score for the Other 18 Students

The total score for the remaining 18 students can be expressed as:

Total score for 18 students Total score for all students - Total score for 34 students

Total score for 18 students 3328 - x times; 34

Step 4: Calculating the Mean Score for the Other 18 Students

Let the mean score for the remaining 18 students be y. Then:

y (Total score for 18 students) / 18

y (3328 - x times; 34) / 18

If we take x 67 for our calculations:

Total score for 34 students 67 times; 34 2278

Total score for 18 students 3328 - 2278 1050

y 1050 / 18 ≈ 58.33

Therefore, the lowest possible mean score for the other 18 students is approximately 58.33.

Conclusion

By understanding and applying the principles of mean scores and weighted averages, we can solve this problem effectively. This problem not only showcases the importance of statistical analysis but also highlights the practical application of mathematical concepts in real-world scenarios.

Additional Insights

It is worth noting that the mean score of a subset of students is dependent on the mean score of the entire class. If the mean score of the subset is maximized, the mean score of the remaining students is minimized. This concept is crucial in various fields, including economics, psychology, and education.