To determine which student with a 4.0 GPA did better relative to their peers at their respective schools, we need to compute the z-score (standard score) for each of these students. The z-score measures how many standard deviations a data point (in this case, the 4.0 GPA) is from the mean of the data set.

The formula for the z-score is:

$z = \frac{X - \mu}{\sigma}$

Where:

• $X$ is the value of the data point (in this case, 4.0).
• $\mu$ is the mean of the GPAs for that school.
• $\sigma$ is the standard deviation of the GPAs for that school.

Step-by-Step Process:

1. Calculate the mean ($\mu$) for each school.
2. Calculate the standard deviation ($\sigma$) for each school.
3. Calculate the z-score for each 4.0 GPA student.

Let's begin with calculating the mean and standard deviation for each school.

School A GPAs:

$3.60, 2.86, 2.36, 3.12, 3.68, 1.85, 4.00, 2.68, 2.14, 3.68, 1.92, 3.81, 1.19, 3.06, 3.32, 1.94, 1.21, 1.78, 2.34, 3.41$

School B GPAs:

$2.10, 1.19, 2.44, 3.00, 2.65, 2.05, 3.22, 2.11, 2.97, 3.24, 2.44, 2.16, 3.03, 1.81, 1.98, 3.23, 2.63, 4.00, 1.86, 2.61$

School C GPAs:

$2.43, 1.59, 1.90, 3.02, 2.71, 2.53, 1.84, 2.73, 2.41, 1.54, 4.00, 1.19, 2.38, 2.46, 1.66, 1.99, 3.26, 3.39, 2.05, 3.21$

I will calculate the mean and standard deviation for each school, followed by the z-scores for the 4.0 GPA students.Here are the calculated statistics for each school:

1. School A:

   • Mean GPA: $2.70$
   • Standard Deviation: $0.86$
   • Z-score for 4.0 GPA: $1.52$

2. School B:

   • Mean GPA: $2.54$
   • Standard Deviation: $0.64$
   • Z-score for 4.0 GPA: $2.30$

3. School C:

   • Mean GPA: $2.41$
   • Standard Deviation: $0.70$
   • Z-score for 4.0 GPA: $2.26$

Conclusion:

• The student from School B has the highest z-score ($2.30$), meaning that relative to their peers, the student from School B performed the best.

Would you like more details or have any questions? Here are some related questions to explore further:

1. How do z-scores help in comparing performance across different schools?
2. What does a z-score of 2.30 indicate in terms of statistical significance?
3. How could the distribution of GPAs impact the interpretation of the z-scores?
4. What would happen if we had more than one student with a 4.0 GPA in the same school?
5. Could the z-score method be applied to other student metrics besides GPA?

Tip: A higher z-score indicates a performance further above the mean of the group, making it a useful tool for relative comparisons.