a) To determine which student did the best, we need to compare their scores relative to their respective groups. We can do this by calculating a z-score for each student, which measures the number of standard deviations above the mean their score falls. The formula for z-score is:
z = (x - μ) / σ
where x is the student's score, μ is the mean score for their grade, and σ is the standard deviation for their grade.
For Jorge, the z-score is:
z = (86.2 - 61.2) / 11.9 = 2.10
For Sophie, the z-score is:
z = (84.3 - 57.9) / 11.6 = 2.28
Since Sophie's z-score is higher, she did better relative to her grade and earned the right to brag.
b) To determine how many students Jorge did better than, we need to find the percentage of students who scored lower than him, and then multiply that percentage by the total number of students. We can use a z-table to find the percentage of students who scored lower than Jorge's z-score of 2.10. The z-table tells us that the area to the left of 2.10 is 0.9821, which means 98.21% of students scored lower than Jorge.
If we assume 10,000 students wrote the math contest, then the number of students Jorge did better than is:
0.9821 * 10,000 = 9,821
Jorge did better than 9,821 students.
c) We can use the same approach as in part (b) to determine how many students did better than Sophie. Her z-score is 2.28, and the area to the left of 2.28 in the z-table is 0.9880, which means 98.80% of students scored lower than Sophie.
If we assume 10,000 students wrote the math contest, then the number of students who did better than Sophie is:
0.9880 * 10,000 = 9,880
9,880 students did better than Sophie.