Answer :
The problem involves analyzing a table of percentages for students signed up for various math classes by grade level. We need to understand what the student incorrectly calculated and what the correct answer is. Here is the detailed, step-by-step solution:
1. Joint Relative Frequency: This term refers to the proportion of students that fall into both of two categories out of the total student population. In this case, it is the percentage of the entire student population that is in 10th grade and is enrolled in Geometry.
2. Conditional Relative Frequency: This term refers to the proportion of students that fall into one category (such as being in 10th grade) given a second category (such as being enrolled in Geometry).
Given the table:
[tex]\[ \begin{tabular}{|l|l|l|l|l|l|} \hline Grade & Geometry & Algebra II & Pre-Calculus & AP Statistics & Total \\ \hline 10 th & 20.7 \% & 10.3 \% & 3.4 \% & 0.7 \% & 35.2 \% \\ \hline 11 th & 6.7 \% & 13.8 \% & 10.3 \% & 2.8 \% & 33.8 \% \\ \hline 12 th & 1.4 \% & 6.7 \% & 13.8 \% & 9 \% & 31 \% \\ \hline Total & 29 \% & 31 \% & 27.6 \% & 12.4 \% & 1 \\ \hline \end{tabular} \][/tex]
### Given:
- The student calculated a percentage of 71.4%.
Let's break down the meaning of these calculations:
- The 71.4% calculation by the student seems to imply that this calculation was made not out of the total student population, but rather out of a specific subset of that population.
- The data shows that of the entire student body, 29% are enrolled in Geometry. It's likely this 71.4% represents Geometry students who are in 10th grade relative to all Geometry students.
### Correct Interpretation:
1. Conditional Relative Frequency: The correct interpretation of 71.4% is that it is the percentage of students in Geometry who are in 10th grade, out of the total students enrolled in Geometry.
[tex]\[ \text{Conditional Relative Frequency} = \frac{\text{10th Grade Geometry Students}}{\text{Total Geometry Students}} = \frac{20.7\%}{29\%} = 71.4\% \][/tex]
2. Joint Relative Frequency: The correct answer for the joint relative frequency of 10th grade students in Geometry is the percentage of all students who are both 10th graders and are enrolled in Geometry, which is given in the table directly.
[tex]\[ \text{Joint Relative Frequency} = 20.7\% \][/tex]
### Conclusion:
The student mistakenly calculated the conditional relative frequency for students who are in 10th grade given that they are enrolled in Geometry. The correct value of the joint relative frequency of 10th grade students in Geometry is 20.7% as per the table.
Therefore, the correct response is:
- The student calculated the conditional relative frequency for students who are in 10th grade, given that they are enrolled in Geometry. The correct value of the joint relative frequency of 10th grade students in Geometry is 20.7%.
1. Joint Relative Frequency: This term refers to the proportion of students that fall into both of two categories out of the total student population. In this case, it is the percentage of the entire student population that is in 10th grade and is enrolled in Geometry.
2. Conditional Relative Frequency: This term refers to the proportion of students that fall into one category (such as being in 10th grade) given a second category (such as being enrolled in Geometry).
Given the table:
[tex]\[ \begin{tabular}{|l|l|l|l|l|l|} \hline Grade & Geometry & Algebra II & Pre-Calculus & AP Statistics & Total \\ \hline 10 th & 20.7 \% & 10.3 \% & 3.4 \% & 0.7 \% & 35.2 \% \\ \hline 11 th & 6.7 \% & 13.8 \% & 10.3 \% & 2.8 \% & 33.8 \% \\ \hline 12 th & 1.4 \% & 6.7 \% & 13.8 \% & 9 \% & 31 \% \\ \hline Total & 29 \% & 31 \% & 27.6 \% & 12.4 \% & 1 \\ \hline \end{tabular} \][/tex]
### Given:
- The student calculated a percentage of 71.4%.
Let's break down the meaning of these calculations:
- The 71.4% calculation by the student seems to imply that this calculation was made not out of the total student population, but rather out of a specific subset of that population.
- The data shows that of the entire student body, 29% are enrolled in Geometry. It's likely this 71.4% represents Geometry students who are in 10th grade relative to all Geometry students.
### Correct Interpretation:
1. Conditional Relative Frequency: The correct interpretation of 71.4% is that it is the percentage of students in Geometry who are in 10th grade, out of the total students enrolled in Geometry.
[tex]\[ \text{Conditional Relative Frequency} = \frac{\text{10th Grade Geometry Students}}{\text{Total Geometry Students}} = \frac{20.7\%}{29\%} = 71.4\% \][/tex]
2. Joint Relative Frequency: The correct answer for the joint relative frequency of 10th grade students in Geometry is the percentage of all students who are both 10th graders and are enrolled in Geometry, which is given in the table directly.
[tex]\[ \text{Joint Relative Frequency} = 20.7\% \][/tex]
### Conclusion:
The student mistakenly calculated the conditional relative frequency for students who are in 10th grade given that they are enrolled in Geometry. The correct value of the joint relative frequency of 10th grade students in Geometry is 20.7% as per the table.
Therefore, the correct response is:
- The student calculated the conditional relative frequency for students who are in 10th grade, given that they are enrolled in Geometry. The correct value of the joint relative frequency of 10th grade students in Geometry is 20.7%.
